Аналитическая геометрия. Часть I: Учебное пособие к курсу

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Author: Шурыгин В.В.

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c defghij kl mnhofhogp p q mihogilkogqe rZr stuvuwXUx txyzuTU {*18-3,69 67:*+36\ 7*|6}69 b -/,-b* ;,-:0N ;*8-L-b 6,67080)*/1-M :*-K ;*8300\ .3*L/86b79*8 /-+-M ~OO*180b,M 0,/832;*,8 L79 3*I*,09 36[K 70),N :*-;*830)*/10N [6L6)= - .-;0;- .30;*,*,09 .30 3*I*,00 [6L6) -,6 ;-|*8 +8? 0/.-7?[-b6,6 L79 .-/83-*,09 :*-;*8300 ,6 -8,-/08*7?,.3-/8-M 61/0-;680)*/1-M +6[*\ -/,-b6,,-M ,6 8*-300 6+/83618,N b*18-3K ,N .3-/836,/8b= { /--8b*8/8b00 / ~80; b ,6/8-9}*; 123/* / /6;-:- ,6)676 36[b0b6*8/9 8*-3*801-K;,-|*/8b*,,M .-LN-L 1 -.3*L*7*,0J :*-;*830)*/10N -+*1K 8-b\ 6 /6;- :*-;*830)*/1-* .3-/836,/8b- 36//;6830b6*8/9 161 ;,-|*/8b-\ ,6L*7*,,-* ,*1-8-3-M /8321823-M\ 1-8-369 0 -.3*L*79*8 *:- :*-;*830)*K /10* /b-M/8b6= {*18-3-; :*-;*830)*/1-:- .3-/836,/8b6 :*-;*830)*/1-M .7-/1-/80 ,6K [b6*8/9 2.-39L-)*,,69 .636 8-)*1 (A, B) ~8-:- .3-/836,/8b6 070 ,6.36bK 7*,,M -83*[-1= -)16 A ,6[b6*8/9 ,6)67-; b*18-36\ 6 8-)16 B 1-,-;= {*18-3 / ,6)67-; b 8-)1* A 0 1-,-; b 8-)1* B .30,98- -+-[,6)68? /7*K → {*18-3 −→ \ 2 1-8-3-:- /-b.6L6J8 ,6)67- 0 1-,*\ L2J}0; -+36[-; −AB = AA ,6[b6*8/9 ,27*b; b*18-3-;= { 8*N /72)69N\ 1-:L6 ,6)67- 0 1-,* b*1K 8-36 ,* [6L6, b 9b,-; b0L* 070 ,* /2}*/8b*,,\ b*18-3 -+-[,6)6J8 .-72|03,;0 +21b6;0 a, b, c, . . . \ 070 +21b6;0 /- /83*71-M ,6b*3N2 ~a \ ~b \ ~c, . . . = → +-[,6K 70,2 -83*[16 AB ,6[b6J8 L70,-M 070 ;-L27*; b*18-36 −AB = → /7*L2J}0; -+36[-; −→ )6*8/9 ;-L27? b*18-36 −AB |AB| = {*18-3 a 0 b \ 36/.-7-|*,,* ,6 .63677*7?,N .39;N\ ,6[b6J8/9 = P-770,*63,-/8? b*18-3-b a 0 b -+-[,6)6*8/9 /7*L2J}0; -+36[-; a||b = 27*b-M b*18-3 /)086*8/9 1-770,*63,; b/91-;2 b*18-32 −→ −−→ AA||CD = P-770,*63,* ,*,27*b* b*18-3 a 0 b ;-:28 +8? ,6.36b7*, b -LK ,2 0 82 |* /8-3-,2 070 b .3-80b-.-7-|,* /8-3-,= 8- -+-[,6)6*8/9\ /--8b*8/8b*,,-\ /7*L2J}0; -+36[-; a ↑↑ b 0 a ↑↓ b = {*18-3 a 0 b ,6[b6J8/9 \ a = b \ */70 |a| = |b| 0 ] a ↑↑ b

.30 |a| = |b| =6 0 = H30 ~8-; b/* ,27*b* b*18-3 36b, ;*|L2 /-+-M= ;**8 ;*/8- /7*L2J}** -)*b0L,-*\ ,- -)*,? b6|,-* TxwuxXWxZ A ¡¢£ ¤¥ a ¡¢¦ → § ¨©ª « ª£ ¤ B £¥ ¤ −AB = a= 36b,N ;*|K 8,-I*,0* 36b*,/8b6 36[+0b6*8 b/* b*18-3 ,6 176// L2 /-+-M b*18-3-b= P6|LM 861-M 176// ,6[b6*8/9 ¬ª = b-+-L,M b*18-3 ;-|,- .3*L/86b798? 161 b*18-3\2 1-8-3-:- [6L6, L70K ,6 0 ,6.36b7*,0*\ ,- ,* [6O01/03-b6,6 ,6)67?,69 8-)16 8-)16 .307-|*K ,09= { L67?,*MI*; +2L*8 .3*L.-768?/9\ )8- b/* 36//;6830b6*;* b*18-3 9b79J8/9 /b-+-L,;0\ 6 b 8-; /72)6*\ 1-:L6 ,2|,- [6O01/03-b68? ,6)67?K ,2J 8-)12 A ,*1-8-3-:- b*18-36 a \ +2L*; :-b-308?\ )8- b*18-3 a -87-|*, -8 8-)10 A = -:76/,- bI*.30b*L*,,-;2 28b*3|L*,0J b/910M b*18-3 /b-K +-L,M® ;-|,- *L0,/8b*,,; /.-/-+-; -87-|08? -8 7J+-M 8-)10= ¯6.0K → {*18-3 −→ .30 ~8-; /b68? ~8- +2L*; 0/.-7?[29 [,61 36b*,/8b6 a = −AB = AB +2L*; ,6[b68? .3*L/86b08*7*; /b-+-L,-:- b*18-36 a = b-+-L,M b*18-3 ) −→ a -L,-[,6 ,- -.3*L*79*8/9 7J+; /b-0; .3*L/86b08*7*; AB 161 176// → b*18-3-b\ 36b,N b*18-32 −AB ,-|*/8b6 /b-+-L,N b*18-3-b :*-;*830K = ° )*/1-:- .3-/836,/8b6 0 :*-;*830)*/1-M .7-/1-/80 +2L*; -+-[,6)68?\ /--8K b*8/8b*,,-\ /0;b-76;0 V 0 V = rZ± ²xyuzuTUx ³uX´zW´ zxuTWW µXux¶ztZ H-,980* ;,-|*/8b6 9b79*8/9 -L,0; 0[ 0/N-L,N .-,980M ;68*;68010= °,-K |*/8b- .3*L/86b79*8 /-+-M /-b-12.,-/8? ,6+-3 -+*18-b 8-M 070 0,-M .303-L\ 1-8-3* ,6[b6J8/9 ~7*;*,86;0 ~8-:- ;,-|*/8b6= 6.30;*3\ ;,-|*/8b- -+36[2J8 b/* 8-)10 .7-/1-/80 070 b*}*/8b*,,* )0/76\ 2L-K b7*8b-39J}0* ,*36b*,/8b2 −1 < x < 1 = 79 ,*1-8-3N )6/8- b/83*)6J}0N/9 ;,-|*/8b .30,98- 0/.-7?[-b68? /86,L638,* -+-[,6)*,09 = 61\ N -[,6)6*8 ;,-|*/8b- b/*N ,682367?,N )0/*7 {1, 2, 3, . . .} \ Z -[,6)6*8 ;,-|*/8b- b/*N *7N )0/*7 {0, ±1, ±2, ±3, . . .} \ ) R · ;,-|*/8b- b/*N b*}*/8b*,,N 0/*7\ C · ;,-|*/8b- b/*N 1-;.7*1/K ,N )0/*7\ 6 C[a; b] · ;,-|*/8b- ,*.3*3b,N O2,10M\ -.3*L*7*,,N ,6 -83*[1* [a; b] = ¸ 3

2

-8 O618\ )8- x 9b79*8/9) ~7*;*,8-; ;,-|*/8b6 M :-b-398 861|*\ )8x .30,6L7*|08 M -+-[,6 6*8/9 /7*L2J}0; -+36[-; x ∈ M 070 M 3 x = ,6[b6*8/9 ¤\ */70 )0/7- ~7*;*,8-b b ~8-; ;,-|*K °,-|*/8b/8b* 1-,*),- = °,-|*/8b- ,6[b6*8/9 ¬¤\ */70 )0/7- ~7*;*,8-b b ~8-; ;,-|*/8b* +*/1-,*),- = ¹-3;67?,- -.3*L*79*8/9 ;,-|*/8b-\ 2 1-K 8-3-:- ,*8 ,0 -L,-:- ~7*;*,86= 61-* ;,-|*/8b- ,6[b6*8/9 ¡ º »= 79 -+-[,6)*,09 .2/8-:- ;,-|*/8b6 0/.-7?[2*8/9 /0;b-7 ∅ = ¼/70 16|LM ~7*;*,8 ;,-|*/8b6 M 9b79*8/9 -L,-b3*;*,,- 0 ~7*;*,K 8-; ;,-|*/8b6 N \ 8- ;,-|*/8b- M ,6[b6*8/9 ¡ª» ;,-|*K /8b6 N = 8- -+-[,6)6*8/9 /7*L2J}0; -+36[-; M ⊂ N 070 N ⊃ M = b6 ;,-|*/8b6 M 0 N /)086J8/9 -L0,61-b;0 070 /-b.6L6J}0;0\ */70 -,0 /-/8-98 0[ -L,0N 0 8*N |* ~7*;*,8-b= 8- -+-[,6)6*8/9 /7*L2J}0; -+36K [-; M = N = )*b0L,-\ M = N 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 M ⊂ N 0 N ⊂ M= [ 0;*J}0N/9 ;,-|*/8b ;-|,- 1-,/83203-b68? ,-b* ;,-|*/8b6= °,-|*/8b-\ /-/8-9}** 0[) b/*N ~7*;*,8-b\ .30,6L7*|6}0N 0 ;,-|*/8b2 M 0 ;,-|*/8b2 N -+}69 6/8? ~80N Lb2N ;,-|*/8b\ ,6[b6*8/9 ¡º ¤ M 0 N = H*3*/*)*,0* ;,-|*/8b M 0 N -+-[,6)6*8/9 /7*L2J}0; -+36[-; M ∩ N = ¼/70 ;,-|*/8b6 M 0 N ,* 0;*J8 -+}0N ~7*;*,8-b\ 0N .*3*/*)*,0* 9b79*8/9 .2/8; ;,-|*/8b-; M ∩ N = ∅ = °,-|*/8b-\ /-/8-9}** 0[ b/*N ~7*;*,8-b\ .30,6L7*|6}0N 70+- ;,-|*K /8b2 M 70+- ;,-|*/8b2 N N-89 + -L,-;2 0[ ~80N Lb2N ;,-|*/8b\ ,6[K b6*8/9 ¬½ª M 0 N = +*L0,*,0* ;,-|*/8b M 0 N -+-[,6)6*8/9 /7*L2J}0; -+36[-; M ∪ N = °,-|*/8b-\ ~7*;*,86;0 1-8-3-:- 9b79J8/9 b/*b-[;-|,* 2.-39L-)*,K ,* .63 (x, y) \ b 1-8-3N .*3bM ~7*;*,8 x .30,6L7*|08 ;,-|*/8b2 M \ 6 b8-3-M ~7*;*,8 y .30,6L7*|08 ;,-|*/8b2 N \ ,6[b6*8/9 ¡¢ª H ) M 0 N = 3-0[b*L*,0* ;,-|*/8b M 0 N -+-[,6 6*8/9 /7*L2J}0; -+36[-; { ) M × N = ~8-; -.3*L*7*,00 .63 (x, y) .3*L.-76:6J8/9 2.-39L- *,,;0 = 8- -[,6)6*8\ )8- (x, y) 6= (y, x) \ */70 x 6= y = P3681- -.3*L*7*,0* .3-0[b*K L*,09 ;,-|*/8b M 0 N [6.0/b6*8/9 /7*L2J}0; -+36[-; M × N = {(x, y) : x ∈ M, y ∈ N }. ¾

5,67-:0),- .3-0[b*L*,0J Lb2N ;,-|*/8b -.3*L*79*8/9 .3-0[b*L*,0* 7J+-:- 1-,*),-:- )0/76 ;,-|*/8b M , M , . . . , M \ n ∈ N ] M × M × . . . × M = {(x , x , . . . , x ) : x ∈ M , i = 1, 2, . . . , n}. { )6/8,-/80\ */70 b/* ;,-|*/8b6 M \ i = 1, 2, . . . , n \ b .3-0[b*L*,00 ] /-b.6L6J8\ 8- */8?\ M = M \ i = 1, 2, . . . , n \ 8- 0N .3-0[b*L*,0* -+-[,6)6K *8/9 161 /8*.*,? M = 6.30;*3\ R = {(x, y) : x ∈ R, y ∈ R} · )0/7-b69 .7-/1-/8?= H2/8? M 0 N · ,*1-8-3* ;,-|*/8b6= ¿¬» ¢ M N ,6[K b6*8/9 /--8b*8/8b0* f \ -8,-/9}** 16|L-;2 ~7*;*,82 x ∈ M ,*1-8-3M ~7*K ;*,8 y ∈ N \ ,6[b6*;M ¬¢ ~7*;*,86 x .30 -8-+36|*,00 f = -\ )8- y 9b79*8/9 -+36[-; ~7*;*,86 x .30 -8-+36|*,00 f \ -+-[,6)6*8/9 /7*L2J}0; -+36[-; y = f (x) 070 x 7→ y = 79 /6;-:- -8-+36|*,09 f .30 ~8-; 0/.-7?K [2J8/9 /7*L2J}0* -+-[,6)*,09 f : M → N 070 f : M 3 x 7→ y ∈ N = 8-+36|*,0* f : M → N ,6[b6*8/9 À½ 070 ¬»º M N \ */70 L79 16|L-:- y ∈ N ,6ML*8/9 N-89 + -L0, ¡¬¢\ 8- */8? 861-M y ∈ N \ )8- y = f (x) = 6.30;*3\ -8-+36|*,09 f : R 3 x 7→ x ∈ R 0 f : R 3 x 7→ 2 ∈ R ,* 9b79J8/9 /J3*180b,;0\ .-/1-7?12 2 -83068*7?,N )0/*7 ,*8 .3--+36[-b\ 6 -8-+36|*,09 f : R 3 x 7→ x ∈ R 0 f : R 3 x 7→ x − x ∈ R 9b79J8/9 /J3*180b,;0\ .-/1-7?12 236b,*,09 x = a 0 x − x = a 0;*J8 3*I*,09 .30 7J+-; a = ªº 8-+36|*,0* f : M → N ,6[b6*8/9 ½ 070 ¢ ¢¤ ¬» M N \ */70 0[ x 6= x /7*L2*8\ )8- f (x ) 6= { ( f (x ) = ;*/8- /7-b6 4/7*L2*8< +2L*; 0/.-7?[-b68? 861|* /0;b-7 ⇒ = /7-K b0* 0,*180b,-/80\ 8610; -+36[-;\ ;-|,- [6.0/68? b b0L* x 6= x ⇒ f (x ) 6= f (x ) = [ 36//;-83*,,N bI* -8-+36|*,0M\ f : R 3 x 7→ x ∈ R 0 f : R 3 x 7→ x − x ∈ R ,* 9b79J8/9 0,*180b,;0\ .-/1-7?12 b .*3b-; /72)6* f (−a) = f (a) L79 7J+-:- a ∈ R \ 6 b- b8-3-; /72K )6* f (−1) = f (0) = f (1) = 0 = 8-+36|*,09 f : R 3 x 7→ x ∈ R 0 f : R 3 x 7→ 2 ∈ R 9b79J8/9 0,*180b,;0 = 8-+36|*,0* f : M → N ,6[b6*8/9 ¬ 070 ¢ªº ¢¤ ¬» M N \ */70 -,- 9b79*8/9 0 /J3*180b,; 0 0,*180b,;\ 8- */8? 2/86,6b70b6*8 b[60;,- -L,-[,6),-* /--8b*8/8b0* Á 1

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;*|L2 ~7*;*,86;0 ;,-|*/8b M 0 N = { ~8-; /72)6* 16|L-;2 ~7*;*,K 82 y ∈ N /--8b*8/8b2*8 -L0, 0 8-7?1- -L0, ~7*;*,8 x ∈ M 861-M\ )8H y = f (x) = 30 ~8-0 -8-+36|*,0* f : M → N \ -8,-/9}** 16|L-;2 ~7*K ;*,82 y ∈ N *:- *L0,/8b*,,M .3--+36[ x ∈ M \ ,6[b6*8/9 ¬ ¬» 1 -8-+36|*,0J f : M → N = [ 36//;-83*,,N bI* .30K ;*3-b\ +0*180b,; 9b79*8/9 8-7?1- -8-+36|*,0* f : R 3 x 7→ x ∈ R = Â0*180b,; 9b79*8/9 861|* -8-+36|*,0* f : R 3 x 7→ 2 ∈ R \ :L* ) R = {x ∈ R : x > 0} · ;,-|*/8b- .-7-|08*7?,N b*}*/8b*,,N 0/*7 = ¡¬¢ Ã a ∈ M .30 -8-+36|*,00 f : M → N ,6[b6*8/9 ;,-|*/8b- f (a) = {x ∈ M : f (x) = a} ⊂ M = ¡º ¬¢ ¡ª» A ⊂ M .30 -8-+36|*,00 f : M → N ,6[b6*8/9 ;,-|*/8b- f (A) = {x ∈ M : f (x) ∈ A} ⊂ M = 6.30;*3\ L79 -8-+36K |*,09 f : R 3 x 7→ x − x ∈ R .-7,M .3--+36[ f (0) /-/8-08 0[ 83*N ~7*;*,8-b f (0) = {−1; 0; 1} \ 6 f (R ) = (−1; 0)∪(1; ∞) · -+*L0,*,0* Lb2N 0,8*3b67-b= Ä » M ,6[b6*8/9 .-L;,-|*/8b- R ⊂ M × M = ¿ ¼/70 (x, y) ∈ R \ 8- :-b-398\ )8- ~7*;*,8 x ,6N-L08/9 b -8,-I*,00 R 1 y = {;*/8- (x, y) ∈ R .0I28 861|* xRy = 8,-I*,0* R ,6 ;,-|*/8b* M ,6[K b6*8/9 Ä Ã\ */70 -,- 2L-b7*8b-39*8 /7*L2J}0; 83*; 2/7-b09; R · 3*O7*1/0b,-\ 8- */8? xRx L79 7J+-:- x ∈ M Å ] R · /0;;*830),-\ 8- */8? xRy ⇒ yRx Å Æ R · 836,[080b,-\ 8- */8? xRy \ yRz ⇒ xRz = )8,-I*,0* ~1b0b67*,8,-/80 )6/8- -+-[,6)6*8/9 /0;b-7-; ∼ = H30 ~8-; 8-\ 8- ~7*;*,8 x 0 y ,6N-L98/9 b -8,-I*,00 ~1b0b67*,8,-/80 ∼ \ -+-K [,6)6*8/9 /7*L2J}0; -+36[-; x ∼ y = °,-|*/8b- [x] = {y ∈ M : y ∼ x} b/*N ~7*;*,8-b 0[ M \ ~1b0b67*,8,N ~7*;*,82 x ∈ M \ ,6[b6*8/9 º Ã Ã x ∈ M = )*b0L,-\ Lb6 176//6 [x] 0 [z] 70+- ,* .*3*/*16J8/9\ 70+- /-b.6L6J8= *M/8b08*7?,-\ */70 w ∈ [x] ∩ [z] 0 y ∈ [z] \ 8- x ∼ w \ w ∼ z 0 z ∼ y = 7*L-b68*7?,-\ x ∼ y \ 0 .-~8-;2 y ∈ [x] = H-/1-7?12 y · .3-0[b-7?,M ~7*;*,8 0[ [z] \ 8- [z] ⊂ [x] = Ç6//2|L69 6,6K 7-:0),; -+36[-;\ .30N-L0; 1 8-;2\ )8- [x] ⊂ [z] = 8/JL6 /7*L2*8\ )8- -8K ,-I*,0* ~1b0b67*,8,-/80 ∼ ,6 ;,-|*/8b* M 36[+0b6*8 ~8- ;,-|*/8b- ,6 È −1

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176//= °,-|*/8b- ~80N 176//-b -+-[,6)6*8/9 /0;b-7-; M/∼ 0 ,6[b6*8K /9 É º» ;,-|*/8b6 M .- -8,-I*,0J ~1b0b67*,8,-/80 ∼ = P6|LM ~7*;*,8 O618-3K;,-|*/8b6 176// ~1b0b67*,8,-/80 [x] ∈ M/∼ -L,-[,6),- -.3*L*79*8/9 7J+; /b-0; ¡ª y ∈ [x] = H-~8-K ;2 L79 [6L6,09 ~7*;*,86 [x] L-/868-),- 216[68? ,*1-8-3M .3*L/86b08*7? ~8-:- 176//6 · ~7*;*,8 ;,-|*/8b6 M = 7*L2J}0* Lb6 .30;*36 .-16[b6J8\ 161 O618-3K;,-|*/8b6 */8*/8b*,K ,; -+36[-; .-9b79J8/9 b :*-;*8300= = ¼/70 Lb6 b*}*/8b*,,N )0/76 α 0 β -870)6J8/9 ,6 )0/7-\ 1368,-* ) ) ) 2π \ 8- [,6 *,09 830:-,-;*830 */10N O2,10M ,6 ~80N 0/76N /-b.6L6J8 H sin α = sin β \ cos α = cos β = -~8-;2 */8*/8b*,,- 36//;6830b68? 63:2;*,8 830:-,-;*830)*/10N O2,10M / 8-),-/8?J L- /76:6*;-:- 2πk = ¼/70 bb*/80 ,6 ;,-|*/8b* b*}*/8b*,,N )0/*7 R /7*L2J}** -8,-I*,0* ~1b0b67*,8,-K /80 α ∼ β ⇔ β − α = 2πk \ 8- O618-3K;,-|*/8b- R/∼ ;-|,- +0*180b,-8-+36[08? ,6 -132|,-/8? *L0,0),-:- 36L02/6 S = {(x, y) ∈ R : x +y = 1} \ .-76:69 f : R/∼ 3 [α] 7→ (cos α; sin α) ∈ S = ]= +-[,6)0; ;,-|*/8b- 8-)*1 :*-;*830)*/1-:- .3-/836,/8b6 /0;b-7-; ) ) ) E = -:L6 ;,-|*/8b- :*-;*830 */10N b*18-3-b 2.-39L- *,,N .63 8- *1 .3*L/86b79*8 /-+-M .3-0[b*L*,0* E ×E = 8,-I*,0* 36b*,/8b6 :*-;*830)*K /10N b*18-3-b 9b79*8/9 -8,-I*,0*; ~1b0b67*,8,-/80 ∼ ,6 E × E = H-~8-K ;2 ;,-|*/8b- /b-+-L,N b*18-3-b .3*L/86b79*8 /-+-M O618-3K;,-|*/8bV = (E × E )/∼ = rZÊ ËWXxÌXUx u³xTÍÎWW XÍw txyzuTÍµW Ï³TxwxxXWxZ ª ª ¬ªÐ a b § ¿» º → a ¦ ¤ A ¥ ¢ −AB = a ¥ ¢ −→ Ñ −→ § Ò¢ ¦ º » b = −BC AB ª − → a + b ¬ ¦ c = AC § Ó*:1- .3-b*308?\ )8- b*18-3 c = a + b ,* [6b0/08 -8 b+-36 ,6)67?,-M 8-)10 A = O-3;2703-b6,,-* .36b07- /7-|*,09 /b-+-L,N b*18-3-b ,6[b6*8/9 1

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¡ ©= ,- b36|6*8/9 /7*L2J}*M O-3;27-M −→ −−→ −→ AB + BC = AC.

8-+36|*,0* + : V × V → V \ -8,-/9}** .63* /b-+-L,N b*18-3-b 0N /2;;2\ {a, b} 7→ a + b \ ,6[b6*8/9 ¡Ñ¦ »£ = 86 -.*3609 -+76L6*8 /7*L2J}0;0 /b-M/8b6;0 1 a + b = b + a 1-;;28680b,-/8? Å 2 (a + b) + c = a + (b + c) 6//-0680b,-/8? Å ) 3 /2}*/8b2*8 b*18-3 0 \ ,6[b6*;M \ 861-M\ 8- L79 7J+-:- b*18-36 a b.-7,9*8/9 a + 0 = a Å ) 4 L79 7J+-:- b*18-36 a /2}*/8b2*8 b*18-3 b 861-M\ 8- a + b = 0 \ ~8-8 b*18-3 -+-[,6)6*8/9 b = −a 0 ,6[b6*8/9 ¡¡» a § -16[68*7?/8b6 /b-M/8b 1 0 2 -)*b0L, 0[ /7*L2J}0N 30/2,1-b 0 ] B Ö a ØÖ b C × Õ ÖØ× b Õ O 3

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3

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a

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A

Ç0/= = Ç0/= ]= [ 30/2,16 7*:1- 2/;6830b6*8/9 ¡ ¡© L79 /7-|*K → L06:-,67? .63677*7-:36;;6 ,09 b*18-3-b /2;;6 a + b = −OC OACB /· → −−→ /8-3-,6;0 −OA = a 0 OB = b = {*18-3 0 · ~8- 176// ,27*bN b*18-3-b\ 6 b*18-3 −a 0;**8 82 |* L70,2\ )8- 0 b*18-3 a \ 0 .3-80b-.-7-|,-* ,6.36b7*,0*\ 8- */8?\ */70 a = −AB → \ 8−→ −a = BA = [ /b-M/8b6 2 /7*L2*8\ )8- 3*[27?868 b)0/7*,09 /2;; n b*18-3-b a + a + . . . + a ,* [6b0/08 -8 36//86,-b10 /1-+-1 .- -.3*L*7*,0J ; ;-|*; /176Lb68? 8-7?1- Lb6 b*18-36®\ 0 L79 ,6N-|L*,09 /2;; a +a + . . . + a ;-|,- b-/.-7?[-b68?/9 /7*L2J}0; ¡ ¢£ ¦ Ù ◦

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−→ -8 ,*1-K -87-|08? b*18-3 a = −AB B 8-3-M 8-)10 A \ [68*; -8 1-,6 b*18-K a −−−→ \ [6K -87-|08? b*18-3 36 BB a a = a 8*; -8 1-,6 b*18-36 a -87-|08? b*1K B 8-3 a = −B−−B→ 0 861 L67** === \ -8 1-,K B a 6 b*18-36 a -87-|08? b*18-3 a = a a −−−−→ B B k = 3, 4, . . . , n − 1 \ 8-:L6 a + −−→ A B Æ a + . . . + a = AB = ; = 30/2,-1 = Ç0/= Æ= [ /b-M/8b6 4 /7*L2*8\ )8- L79 7J+N b*1K 8-3-b a 0 b 236b,*,0* a + x = b 0;**8 *L0,K b−a /8b*,,-* 3*I*,0* x = b + (−a) \ ~8-8 b*1K A B 8-3 ,6[b6*8/9 ¢À b*18-3-b b 0 a 0 -+-[,6)6*8/9 x = b − a = [ .36b076 /7-K b a |*,09 b*18-3-b b8*16*8 /7*L2J}** .36b07O L79 ,6N-|L*,09 36[,-/80 b − a */70 -87-K →\ Ç0/= Ú = |08? b*18-3 a 0 b -8 -L,-M 8-)10 a = −OA −−→ −→ b = OB \ 8- b − a = AB = ; = 30/2,-1 Ú = Ï³TxwxxXWxZ ¢ª «© ¤ λ a º ¢£ λa ¥ ª¢¤ ¡ª£¦ ªÀ« £Û 1) |λa| = |λ||a| ¥ 2) λa ↑↑ a ¥ λ > 0 ¥ a 6= 0 ¥ λa ↑↓ a ¥ λ < 0 ¥ a 6= 0 § 8-+36|*,0* · : R×V → V \ -8,-/9}** .63*\ /-/8-9}*M 0[ b*}*/8b*,K ,-:- )0/76 0 b*18-36\ 0N .3-0[b*L*,0*\ {λ, a} 7→ λ · a = λa \ ,6[b6*8/9 ¡Ñ¦ »£ ¤= * 832L,- .3-b*308?\ )8- ~86 -.*3609 -+76L6*8 /b-M/8b6;0 5 λ(a + b) = λa + λb L0/830+280b,-/8? -8,-/08*7?,- /7-|*,09 b*1K 8-3-bÅ ) 6 (λ + µ)a = λa + µa L0/830+280b,-/8? -8,-/08*7?,- /7-|*,09 0K /*7Å 7 (λµ)a = λ(µa) 8 1 · a = a\ 1

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1-8-3* ; bL*70; -/-+-\ 6 861|* /7*L2J}0;0 /b-M/8b6;0 0 · a = 0,

λ · 0 = 0,

(−1) · a = −a,

Æ

1-8-3* ;-|,- bb*/80 O-3;67?,- 0[ /b-M/8b 5 · 8 = [ -.3*L*7*,09 -.*3600 2;,-|*,09 )0/76 ,6 b*18-3 b8*16*8\ 13-;* 8-:-\ /7*L2J}** /b-M/8b- 1-770,*63,N b*18-3-b TxwuxXWxZ Þ b||a a 6= 0 ¥ « ª ¤ λ ∈ R ¥ ¤ b = λa = *M/8b08*7?,-\ b = a .30 a ↑↑ b 0 b = − a .30 a ↑↓ b = Ï³TxwxxXWxZ Þ a 6= 0 b = λa ¥ ¤ λ ¢£ Äº Ð b a § ß Ä ¬¢¤£ ªº À« ¬¢Û λ = = àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*100 \ ]Å âä \ :76b6 @\ å= æ hhiçpklgj qefghihq ' ±Zr YxyzuTXux ³Tu¶zTÍX¶ztuZ H-,980* b*18-36 -16[b6*8/9 -)*,? 2L-+,; L79 61/0-;680[600 :*-;*8K 300= Ç6//;-83*,,* bI* /b-M/8b6 1 · 8 -.*360M /7-|*,09 b*18-3-b 0 2;,-|*,09 b*18-36 ,6 )0/7- .30b-L98 1 /7*L2J}*;2 -.3*L*7*,0J Ï³TxwxxXWxZ è ¡ ( ª ¡ «Ð ¤) ¢£ ¦ (V, +, · ) ¥ £«£ ¢ © »º V ªÐ ¬»¦Û + : V × V → V ¥ £«© ¡ Ãº a, b ¢ V ¦ ¦ Ã Ã© »¥ {a, b} 7→ a + b ¥ ¢¦ ¦ a b ¥ · : R × V → V ¥ £«© ¡º ¥ £«¦ ¢ «© ¤ λ Ã a » V ¥ ¦ Ã Ã© »¥ {λ, a} 7→ λ · a = λa ¥ ¢¦ ¡¢ª λ a ¥ ¬ªÀ Ä¢ ¦ 1 8 § 7*;*,8 ;,-|*/8b6 V .30 ~8-; ,6[b6J8 b 6+/83618,-; /;/7* /7-b6= 79 .3-/8-8 b*18-3,-* .3-/836,/8b- (V, +, · ) -+-[,6)6J8 -L,0; /0;b-7-; V = ÜÜ ◦

|b| |a|

|b| |a|

b a

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610; -+36[-;\/b-M/8b6 1 · 8 9b79J8/9 61/0-;6;0 b 6+/83618,-; b*1K 8-3,-; .3-/836,/8b* V = H-;0;- /b-M/8b 1 · 8 \ 1-8-3;0 -.*3600 /7-|*,09 b*18-3-b 0 2;,-K |*,09 b*18-36 ,6 )0/7- b 6+/83618,-; b*18-3,-; .3-/836,/8b* V -+76L6K J8 .- -.3*L*7*,0J\ ~80 -.*3600 -+76L6J8 0 L32:0;0 /b-M/8b6;0 -.*36K 0M /7-|*,09 b*18-3-b 0 2;,-|*,09 b*18-36 ,6 )0/7- b :*-;*830)*/1-; .3-/836,/8b* = - ~80 L32:0* /b-M/8b6 b 6+/83618,-; b*18-3,-; .3-/8K 36,/8b* b8*16J8 0[ /b-M/8b 1 · 8 \ 8- */8? 9b79J8/9 O-3;67?,;0 /7*LK /8b09;0 /b-M/8b 1 · 8 = TxwuxXWxZ è ¡ V £ ª º ¦ 0 ¥ ª 0 0 ª£À 3 ¥ 0 = 0 § éuyÍêÍzxë¶ztuZ 61 161 16|LM 0[ b*18-3-b 0 0 0 2L-b7*8b-39*8 61/0-;* 3 \ 6 /7-|*,0* b*18-3-b 1-;;28680b,- 61/0-;6 1 \ 8◦

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01 = 01 + 02 = 02 .

TxwuxXWxZ è ¡ V £ ª −a ¡¡»¦ a ¥ ª (−a) (−a) ª£À 4 ¥ (−a) = (−a) § éuyÍêÍzxë¶ztuZ 61 161 16|LM 0[ b*18-3-b (−a) 0 (−a) 2L-b7*8b-K 39*8 61/0-;* 4 \ 6 /7-|*,0* b*18-3-b 6//-0680b,- 61/0-;6 2 \ 81

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1

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(−a)1 = (−a)1 + 0 = (−a)1 + (a + (−a)2 ) = = ((−a)1 + a) + (−a)2 = 0 + (−a)2 = (−a)2 .

[ 61/0-; 4 /7*L2*8\ )8- 161 0 b /72)6* :*-;*830)*/1-:- .3-/836,K /8b6 L79 7J+N b*18-3-b a 0 b b*18-3,-:- .3-/836,/8b6 V 236b,*,0* a + x = b 0;**8 *L0,/8b*,,-* 3*I*,0* x = b + (−a) = TxwuxXWxZ ¿¡Ñ£ »£ ¤ ìíî¥ ª£ À¬© a ∈ V ¡ V ¬ª ¦ À¬© λ ∈ R À ªÀ« Û Ú 0 · a = 0, λ · 0 = 0, (−1) · a = −a. ◦

Üï

éuyÍêÍzxë¶ztuZ H*3b-* 36b*,/8b- Ú /7*L2*8 0[ 61/0-; 8 \ 6 0 *L0,K /8b*,,-/80 3*I*,09 x = 0 236b,*,09 a + x = a ◦

◦

a = 1 · a = (1 + 0)a = 1 · a + 0 · a = a + 0 · a.

{8-3-* 36b*,/8b- Ú /7*L2*8 0[ 61/0-; 7 0 .*3b-:- 36b*,/8b6 Ú ◦

λ · 0 = λ(0 · a) = (λ · 0)a = 0 · a = 0.

3*8?* 36b*,/8b- Ú L-16[b6*8/9 /7*L2J}0; -+36[-; (−1) · a + a = (−1) · a + 1 · a = ((−1) + 1) · a = 0 · a ⇒ (−1) · a = −a.

ðÍwÍñÍ rZ -16|08*\ 0/.-7?[29 ;*8-L ;68*;680)*/1-M 0,L2100\ )83*[27?868 b)0/7*,09 /2;; n b*18-3-b ò a + a + ... + a b b*18-3,-; .3-/836,/8b* V ,* [6b0/08 -8 36//86,-b10 /1-+-1= óyÍêÍXWxZ /,-b6,0*; 0,L2100 .30 n = 3 9b79*8/9 61/0-;6 2 6/K /-0680b,-/80 /7-|*,09 b*18-3-b= H*3*N-L -8 /2;; k b*18-3-b 1 /2;;* { k + 1 b*18-36 -/2}*/8b79*8/9 /7*L2J}0; -+36[-; = [6b0/0;-/80 -8 .-K 39L16 -/2}*/8b7*,09 L*M/8b0M /7-|*,09 Lb2N b*18-3-b\ b/*:- 0;**8/9 ) H ò k! /.-/-+-b b 0/7*,09 /2;; .30 n = k + 1 = - 8-;2\ 161-* L*MK /8b0* -/2}*/8b79*8/9 .*3b;\ b/* ~80 /.-/-+ 36[+0b6J8/9 ,6 k 176//-b H C , . . . , C b 176//* C .*3b-* L*M/8b0* · /7-|*,0* a +a = - .3*L.-7-K |*,0J 0,L2100 b/* /.-/-+ b)0/7*,09 /2;;\ .30,6L7*|6}0* -L,-;2 176//2\ L6J8 -L0, 0 8-8 |* 3*[27?868= - .- .3*L.-7-|*,0J 0,L2100 -L0, 0 8-8 |* 3*[27?868 L6J8 861|* 7J+* Lb6 /.-/-+6\ ,6)0,6J}0*/9 / (a + a ) + a 0 / a + (a + a ) \ 8- */8? 7J+* Lb6 /.-/-+6 0[ 176//-b .30b-L98 1 -L,-;2 3*[27?8682= C 0C ±Z± ËWXxÌXÍ´ êÍtW¶Wµu¶zë txyzuTutZ H3-L-7|6*; 36//;6830b68? .3-0[b-7?,-* b*18-3,-* .3-/836,/8b- V = Ï³TxwxxXW´Z ô¦¦ ¬Ñ¦ a , a , . . . , a ÃÉÉº Ñ λ , λ , . . . , λ ¢£ 1

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λ1 a1 + λ2 a2 + . . . + λk ak .

Ü

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ô¦£ ¬Ñ£¥ ÃÉÉÑ ¦ ÀÛ λ = λ = ª · · · = λ = 0 ¥ ¢£ ¦§ ¿¤ ¥ £ ¦£ ¬Ñ£ À¬Ð a , a , . . . , a £££ 0 § è a , a , . . . , a ¢À£ ¦ ¢¥ «º £ ¦£ ¬Ñ£ ÃÐ £ º Û 1

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λ1 a1 + λ2 a2 + . . . + λk ak = 0.

Þ » £ ¦£ ¬º Ñ£ a , a , . . . , a ¥ Ã ¢À£ ¦ ¢º § stuÌ¶ztÍ WXxÌXu êÍtW¶WµUõ ¶W¶zxµ txyzuTutZ è a , a , . . . , a ¦ ¢ ©ª ©ª¥ º ©ª ª ¢ Ð £££ ¦¦ ¬Ñ¦ ª©Ð§ *M/8b08*7?,-\ */70 a = λ a +. . .+λ a \8- (−1)a +λ a +. . .+λ a = 0= +368,-\ */70 λ a + λ a + . . . + λ a = 0 0\ ,6.30;*3\ λ 6= 0 \ 8a = −(λ /λ )a − . . . − (λ /λ )a = ] Þ ª a , a , . . . , a £ ¦¥ Ã ££À£ ¦ ¢§ -16[68*7?/8b- ~8-:- /b-M/8b6 -)*b0L,- */70\ ,6.30;*3\ a = 0 \ 81 · a + 0 · a + . . . + 0 · a = 0 · ,*830b067?,69 70,*M,69 1-;+0,609 = Æ Þ ¦ ¢ a , a , . . . , a ª¬ À¬ b , b , . . . , b ¥ a , a , . . . , a ¥ b , b , . . . , b ¬ª ¦ ¢¦§ *M/8b08*7?,-\ */70 λ a + λ a + . . . + λ a = 0 · ,*830b067?,69 70,*M,69 1-;+0,609\ 8- 0 λ a + λ a + . . . + λ a + 0 · b + . . . + 0 · b = 0 70,*M,69 1-;+0,609 = · ,*830b067?,69 Ú ¿ª a ¦ ¢ ©ª ©ª¥ ©ª ¦Û a = 0 § -16[68*7?/8b- ~8-:- /b-M/8b6 -)*b0L,- = { .3-/836,/8b* V /b-+-L,N b*18-3-b :*-;*830)*/1-:- .3-/836,/8b6\ 13-;* 8-:-\ b.-7,9J8/9 /7*L2J}0* /b-M/8b6 1

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ò ö a , a ¦ ¢ ©ª ©ª¥ ©ª Û a ||a § _ ¨ a , a , a ¦ ¢ ©ª ©ª¥ ©ª ¡ ª¦ ¡§ 610* b*18-3 ,6[b6J8/9 ¡= 27*b-M b*18-3 /)086*8/9 .63677*7?,; b/91-M .7-/1-/80= ã ôÀ¬ ¤ a , a , a , a ¦ ¢§ H30 L-16[6K 8*7?/8b* ~8-:- /b-M/8b6 ;-|,- /)0868?\ )8- b*18-3 a \ a \ a · 70,*M,,*[6b0/0; b /72)6*\ 1-:L6 -,0 70,*M,- [6b0/0;\ 28b*3|L*,0* -)*b0LK −→ \ ,-= 87-|0; b/* )*83* b*18-36 -8 -L,-M 8-)10 O = H2/8? a = −OA ) i = 1, 2, 3, 4 = .3-*18032*; [68*; 8- 12 A ,6 16|L2J 0[ .39;N OA \ H OA 0 OA .63677*7?,- Lb2; L32:0; = 2/8? .3-*109;0 9b79J8/9\ /-K −→ 0 −−→ 1-770,*K -8b*8/8b*,,-\ 8-)10 B \ B 0 B = 61 161 b*18-3 −OB OA −→ −−→ 63,\ 8- ,6ML28/9 8610* )0/76 λ \ i = 1, 2, 3 \ )8- −OB = λ OA = -:L6 a =λ a +λ a +λ a = TxwuxXWx ÷Ï¶XutXÍ´ xµµÍ u WXxÌXuÌ êÍtW¶Wµu¶zW txyzuTutøZ ¡ V ¢ª ª ¬ {a , a , . . . , a } {b , b , . . . , b } ¥ ¡ ¡ Ã »ª¦ ¡© ¬ º ¦ »£ ¤¢ © ¬¥ a = λ b + ª λ b + . . . + λ b ¥ α = 1, 2, . . . , n ¥ ¥ ¡ ¬¦ ¢¡¥ 1

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  a1 = λ11 b1 + λ21 b2 + . . . + λm  1 bm    a = λ1 b + λ2 b + . . . + λm b 2 2 1 2 2 2 m  ...     an = λ1 b1 + λ2 b2 + . . . + λm bm . n n n

_

¨©ª ¡ n > m {a , a , . . . , a } ¦ ¢= éuyÍêÍzxë¶ztuZ H30;*,0; ;*8-L ;68*;680)*/1-M 0,L2100= H30 m = 1 0;**; a = λ b \ α = 1, 2, . . . , n = ¼/70 a = 0 \ 8- b*18-3 ¼ a 70,*M,- [6b0/0; .- /b-M/8b2 ] = /70 a 6= 0 \ 8- λ a − λ a = 0 = H3*L.-7-|0; 8*.*3?\ )8- 28b*3|L*,0* /.36b*L70b- .30 b/*N m 6 k \ 0 L-16|*;\ )8- -,- /.36b*L70b- 0 .30 m = k + 1 = H2/8? 0;**8 ;*/8,6+-3 70,*M,N 1-;+0,60M _\ :L* m = k + 1 0 n > m = ¼/70 b/* 1-K ~OO00*,8 .30 b*18-3* b b 70,*M,N 1-;+0,609N _ 36b, ,27J Ü¾ 1

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1 2 1

\ 8- b*18-3 a \ α = 1, 2, . . . , n \ 9b79J8/9 70K ,*M,;0 1-;+0,609;0 b*18-3-b b , . . . , b 0\ /7*L-b68*7?,-\ 70,*M,[6b0/0; .- .3*L.-7-|*,0J 0,L2100= H2/8? 8*.*3? /3*L0 1-~OO00*,K 8-b λ b 1-;+0,609N _ 0;**8/9 ,*,27*b-M= °-|,- /)0868?\ )8- λ 6= 0 b .3-80b,-; /72)6* .*3*,2;*32*; b*18-3 a , . . . , a 8610; -+36[-;\ )8-+ b.-7,97-/? λ 6= 0= Ç6//;-830; ,6+-3 b*18-3-b λ ã c = a − a , α = 2, . . . , n. λ H-L/86b799 b O-3;27 ã 70,*M,* 1-;+0,600 _\ .-72)0; .3*L/86b7*K ,09 b*18-3-b c , . . . , c b b0L* 70,*M,N 1-;+0,60M b*18-3-b b , . . . , b

λ11 = λ12 = . . . = λ1k+1 = 0 k

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 k+1 2   c2 = µ2 b2 + . . . + µ2 bk+1 ...   cn = µ2n b2 + . . . + µk+1 n bk+1 .

k+1

a

H-/1-7?12 ,6+-3 {c , . . . , c } /-/8-08 0[ (n−1)K:- b*18-36\6 ,6+-3 {b , . . . , b /-/8-08 0[ k b*18-3-b\ 0 n−1 > k \8- .- .3*L.-7-|*,0J 0,L2100 b*18-3 c , . . . , c 9b79J8/9 70,*M,- [6b0/0;;0\ 8- */8? 0;**8/9 ,*830b067?,69 70,*M,69 1-;+0,609 b*18-3-b ~8-:- ,6+-36\ 36b,69 ,27J ` ν c + . . . + ν c = 0. H-L/86b799 b ` b36|*,09 ã L79 b*18-3-b c \ .-72)0; 36b*,/8bλ λ ^ ν (a − a ) + . . . + ν (a − a ) = 0. λ λ Ç6/13b69 /1-+10 b 7*b-M )6/80 36b*,/8b6 ^\ .-72)0; ,*830b067?K ,2J 70,*M,2J 1-;+0,60J b*18-3-b a \ α = 1, 2, . . . , n = H-L3-+,-* 36//;-83*,0* /b-M/8b /0/8*; b*18-3-b ;-|,- ,6M80 b 3*1-K ;*,L2;-M ,0|* 708*36823* = àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*109 ]Å âä \ :76b6 ù@\ åÅ âòä \ :76b6 @@\ åÅ âÆä \ :76b6 @= Ï³TxwxxXWxZ è ¡ V ¢£ n º ¡º ¢ n¥ Ã ¡ ¡£À£ ª ªÀ« Û ÜÁ 2

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2

k+1 }

ú«À n ¦ ¢Ð § ôÀ¬¦ ¬ ¢ n + 1 £££ ¦ ¢§ >0/7- n ,6[b6*8/9 ¢À .3-/836,/8b6 V 0 -+-[,6)6*8/9 /7*K L2J}0; -+36[-; n = dim V = ¼/70 L79 7J+-:- n b b*18-3,-; .3-/836,/8b* V /2}*/8b2J8 n 70,*M,,*[6b0/0;N b*18-3-b\ ~8- .3-/836,/8b- ,6[b6*8/9 ¬¤= {*18-3,-* .3-/836,/8b-\ 0;*J}** 1-,*),2J 36[;*3,-/8?\,6[b6J8 861|* ¤= 79 -+-[,6)*,09 b*18-3,-:- .3-/836,/8b6 36[;*3,-/80 n .30,98- 0/.-7?[-b68? /0;b-7 V = -+36[-;\ /b-+-L,* b*18-3 :*-;*830)*/1-:- .3-/836,/8b6 -+36K 610; [2J8 ÆK;*3,-* b*18-3,-* .3-/836,/8b-\ /b-+-L,* b*18-3 :*-;*830)*/1-M .7-/1-/80 · ]K;*3,-* b*18-3,-* .3-/836,/8b-\ 6 /b-+-L,* b*18-3 .39K ;-M · K;*3,-* b*18-3,-* .3-/836,/8b- = H30;*3-; +*/1-,*),-;*3,-:- b*18-3,-:- .3-/836,/8b6 ;-|*8 /72|08? ;,-|*/8b- C[a; b] ,*.3*3b,N O2,10M\ -.3*L*7*,,N ,6 -83*[1* [a; b] \ / -+),;0 -.*3609;0 /7-|*,09 O2,10M 0 2;,-|*,09 O2,100 ,6 )0/7- = H30 7J+-; n ∈ N ;,-:-)7*, x, x , . . . , x -+36[2J8 70,*M,- ,*[6b0/0;2J /0/8*;2= Ï³TxwxxXWxZ V ¡§ ª» ª ª W ⊂ V ¢£ ¡ ¡ V ¥ £ À¬Ð a, b ∈ W ª£ À¬© λ ∈ R ¡££Û 9◦ 10◦

n

2

a + b ∈ W,

n

λa ∈ W.

[ ~8-:- -.3*L*7*,09 /7*L2*8\ )8- -.*3600 +:V×V →V 0 ·:R×V →V 0,L2032J8 -8-+36|*,09 +:W×W →W 0 ·:R×W →W 8-),** +7- + -+-[,6)08? ~80 -8-+36|*,09 + 0 · \ 0 83-M16 (W, +, ·) 9b79*8/9 b*18-3,; .3-/836,/8b-;= ¼/70 dim V = n \ 8-\ 161 ~8- /7*L2*8 0[ /b-M/8b 70,*M,- [6b0/0;N /0/8*;\ dim W 6 dim V = ÜÈ W

W

H2/8? π · ,*1-8-369 .7-/1-/8?\ 6 ` · ,*1-8-369 .39;69 b :*-;*830)*K /1-; .3-/836,/8b*\ 8-:L6 b*18-3 .3-/836,/8b6\ .63677*7?,* .7-/1-/80 π \-+36[2J8 Lb2;*3,-* .-L.3-/836,/8b- V (π) ⊂ V \6 b*18-3 .63677*7?K ,* ` \ · -L,-;*3,-* .-L.3-/836,/8b- V (`) ⊂ V = 80 .-L.3-/836,/8b6 ,6[b6J8/9 ¡£À« ¡ª¡ .7-/1-/80 π 0 .39;-M ` /--8b*8/8b*,,- = ±ZÊ ûÍêW¶ t txyzuTXuµ ³Tu¶zTÍX¶ztxZ Ï³TxwxxXWxZ ü¢ n º ¡ V ¢º £ ¡£ª¤¦ ¬ {e , e , . . . , e } = {e } ¥ i = 1, . . . , n ¥ ¢ n º ¦ ¢Ð § ÓJ+-M b*18-3 a ∈ V -+36[2*8 / b*18-36;0 +6[0/6 /0/8*;2\ /-/8-9}2J 0[ n + 1 b*18-36\ 1-8-369 9b79*8/9 70,*M,- [6b0/0;-M .- 61/0-;* 10 = 8/JL6 /7*L2*8\ )8- /2}*/8b2*8 ,*830b067?,69 70,*M,69 1-;+0,609 2

3

1

3

n

1

2

n

i

n

◦

λ1 e1 + λ2 e2 + . . . + λn en + λa = 0.

{ ~8-M 70,*M,-M 1-;+0,600 λ ,* ;-|*8 36b,98?/9 ,27J .-)*;2ý= H-L*K 70b ** ,6 λ \ .3*L/86b0; b*18-3 a b b0L* X ae. a = a e + a e + ... + a e = [ -/,-b,-M 7*;; - 70,*M,-M [6b0/0;-/80 /7*L2*8\ )8- b nK;*3,-; b*18-3,-; .3-/836,/8b* V ,* ;-|*8 /2}*/8b-b68? ,6+-36 0[ m < n b*1K 8-3-b\ )*3*[ 1-8-3* 70,*M,- b36|6*8/9 7J+-M b*18-3 0[ V = ðÍµxñÍXWx ÷TÍtWu ¶SµµWTutÍXW´ þÌXÿzxÌXÍøZ { :*-;*8300 0 70,*MK ,-M 67:*+3* .30,98- /2;; b0L6 [6.0/b68?\ -.2/169 [,61 /2;;03-K b6,09 a = a e = { L67?,*MI*; b/JL2 b b36|*,09N b0L6 n

1

2

1

n

2

i

n

i

i=1

n

n

i

i

ai ei ,

ai bi ,

gij ai bj

.- 16|L-M .63* -L0,61-bN 0,L*1/-b\ 0[ 1-8-3N -L0, /8-08 b,0[2\ 6 L32K :-M ,6b*3N2\ .3*L.-76:6*8/9 /2;;03-b6,0* .- b/*M 0N -+76/80 0[;*,*,09 = P-~OO00*,8 {a , a , . . . , a } = {a } 36[7-|*,09 b*18-36 a .- +6K [0/2 {e } ,6[b6J8/9 ª a b +6[0/* 070 -8,-/08*7?,+6[0/6 {e } = ÜÔ 1

i

i

2

n

i

TxwuxXWxZ ª ª© ¬¢ ¡º ª£À£ ª¢¤= */8b08*7?,-\ .3*L.-7-|0b\ )8- a = a e = b e \ .-72)0; a e −b e = 0 \ -812L6 (a − b )e = 0 0\ /7*L-b68*7?,-\ 861 161 b*18-3 +6[0/6 ,* ;-:28 +8? 70,*M,- [6b0/0;;0\ a − b = 0 .30 b/*N i = 1, . . . , n = 7*L2J}** 28b*3|L*,0* ,*;*L7*,,- /7*L2*8 0[ -L,-[,6),-M -.3*L*7*,K ,-/80 1--3L0,68 b*18-3-b TxwuxXWxZ À ªÀ« Ä£ i

i

i

i

i

i

i

i

i

i

i

i

i

1) c = a + b ⇐⇒ ci = ai + bi , i = 1, . . . , n; 2) a = λb ⇐⇒ ai = λbi , i = 1, . . . , n; 3) b = λ1 a1 +λ2 a2 +. . .+λk ak ⇐⇒ bi = λ1 ai1 +λ2 ai2 +. . .+λk aik , i = 1, . . . , n

= 79 16|L-:- ,682367?,-:- )0/76 n 0;**8/9 4/86,L638,-*< b*18-3,-* .3-/836,/8b- 36[;*3,-/80 n = 8- .3-/836,/8b- R \ ~7*;*,86;0 1-8-3-:9b79J8/9 nK/83-10 {a } = {a , . . . , a } \ /-/8-9}0* 0[ b*}*/8b*,,N )0/*7\ /- /7*L2J}0;0 -.*3609;0 /7-|*,09 0 2;,-|*,09 ,6 )0/7- n

i

1

n

{ai } + {bi } = {ai + bi },

λ{ai } = {λai }.

86,L638,M +6[0/ b ~8-; .3-/836,/8b* -+36[2J8 /7*L2J}0* /83-10 {e = {0, . . . , 1, . . . , 0}} \ :L* b /83-1* e *L0,06 /8-08 ,6 iK8-; ;*/8*\ 6 -/867?,* ~7*;*,8 ~8-M /83-10 ,27*b* = H30 ~8-; {a } = a e = Ï³TxwxxXWxZ V W ¡§ ¿¬»º ϕ : V → W ¢£ ¦¥ ª£ ªÀº « ª £Û i

i

i

1) ϕ(a + b) = ϕ(a) + ϕ(b);

ϕ(λa + µb) ϕ − 7 −−→

Õ

i

2) ϕ(λa) = λϕ(a).

λa + µb

Ö

i

Ç0/= ò= ÜÙ

ϕ(b)

ϕ(a)

ðÍµxñÍXWxZ )*b0L,-\ -8-+36|*,0* ϕ : V → W 9b79*8/9 70,*M,; 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 L79 7J+-M 70,*M,-M 1-;+0,600 b = λ a + λ a + . . . + λ a b.-7,9*8/9 ϕ(b) = λ ϕ(a ) + λ ϕ(a ) + . . . + λ ϕ(a ) = H-7* b*}*/8b*,,N )0/*7 R / -.*3609;0 /7-|*,09 0 2;,-|*,09 )0/*7 9b79*8/9 K;*3,; b*18-3,; .3-/836,/8b-;= P61 /7*L/8b0* 0[ L-16[6,K ,N bI* .3*L7-|*,0M\ .-72)6*; TxwuxXWxZ {e } ¥ i = 1, . . . , n ¥ ¬¢ ¡º V § ¨©ª ¬» e : V 3 a 7→ a ∈ R ¥ £« a © i ºÀ ª a Ã© ¬¢¥ £££ ¦ ¬»= 1

2

k

2

1

k

2

1

2

k

1

k

i

i

n

i

n

i

a e

a2

2

b

e2

b2

^

Ç0/= _= TxwuxXWxZ V ¥ V V ¡¥ ϕ : ª V → V ψ : V → V ¦ ¬»£§ ¨© ¡¢Ñ£ ψ ◦ ϕ : V → V » £££ ¦ ¬»= *M/8b08*7?,-\ .*3b-* 2/7-b0* 70,*M,-/80\ ,6.30;*3\ .3-b*39*8/9 /7*K L2J}0; -+36[-; (ψ ◦ ϕ)(a + b) = ψ(ϕ(a + b)) = ψ(ϕ(a) + ϕ(b)) = ψ(ϕ(a)) + ψ(ϕ(b)) = (ψ ◦ ϕ)(a) + (ψ ◦ ϕ)(b) = Ï³TxwxxXWxZ è¢ª¢¤ ¦ ¬» ϕ : V → W ¢£ ¢É¢§ Þ « ¢É¢ ϕ : V → W ¥ ¡ V W ¢À£ ¢É§ ;**8 ;*/8- /7*L2J}** -)*b0L,-* TxwuxXWxZ ϕ : V → V ψ : V → V ¢É¢ Ð ¡§ ¨©ª ¬ ¬» ϕ : V → V ¡¢Ñ£ ψ ◦ ϕ : V → V ¢É¢ Ð ¡= ïÝ e1

0

0

0

00

00

00

0

0

00

−1

00

0

] ¿¬» {e } : V 3 a 7→ {a } = {a , . . . , a } ∈ R ¥ £« a ¬ © ª {a } © ¬¢ {e } ¥ ¢É¢= sxw¶ztWxZ ôÀ¬ ª Ð ¡ ª¦ ¢ V W ¢É= *M/8b08*7?,-\ 16|L-* 0[ ,0N 0[-;-3O,- .3-/836,/8b2 R = /2}*/8b2*8 8-7?1- -L,610; -+36[-;\ / 8-),-/8?J L- 0[-;-3O0[;6\ b*18-3,-* .3-/836,/8b- 36[;*3,-/80 n = { )6/8,-/80\ b/* b*18-3,* .3-/8K 36,/8b6 36[;*3,-/80 3 0[-;-3O, .3-/836,/8b2 /b-+-L,N b*18-3-b :*-K ;*830)*/1-:- .3-/836,/8b6 0 b*18-3,-;2 .3-/836,/8b2 R = { :*-;*830)*/1-; .3-/836,/8b* +6[0/ -+36[2J8 7J+* 830 ,*1-;.76K ,63,N b*18-36 {e , e , e } \ 6 ,6 .7-/1-/80 · 7J+* Lb6 ,*1-770,*63,N b*18-36 {e , e } = H3-0[b-7?,M b*18-3 a .30 ~8-; 0;**8\ /--8b*8/8b*,,-\ b0L a = a e + a e + a e 0 a = a e + a e = i

1

i

n

n

n

i

i

n

n

n

3

1

1

1

2

3

2

2

1

3

2

1

3

1

2

2

x3

x2 a

e3 e1

a

e2 x1

x2

e2

e1

x1

Ç0/= ã=

±Z TWµxTUZ → 9b79*8/9\ b /-K /,-b,-M 0L**M .30 -8/16,00 ,*0[b*/8,-:- b*18-36 −AB −→ \ -8b*8/8b00 / .36b07-; [6;16,09 7-;6,-M\ ,6N-|L*,0* *.-)10 −AM −−−→ \ −−−→ \ −−−→ −→ M M M M . . . \ M B \ /-/8-9}*M 0[ 0[b*/8,N b*18-3-b = -:L6 AB = −−→ −−−→ −−−→ −−−→ AM + M M + M M + . . . + M B = ðÍwÍñÍ ±Z { 83*2:-7?,01* ABC AF · ;*L06,6\ 6 E · *,83 89|*/80 → )*3*[ −→ 0 −→ 8-)16 .*3*/*)*,09 ;*L06,= {36[08? b*18-3 −AF→ 0 −AE AB AC = 1

1

2

1

2

3

1

2

k

2

3

k

ïÜ

àxÿxXWxZ ;= 30/2,-1 a= ;**; −AF→ = −AC → −→ + CF 1 −→ −→ (AB − AC) = 2 −→ AE =

,6N-L0;

C

−→ = AC +

= 8/JL6 =

1 −→ −→ (AB + AC) 2 1 −→ −→ 2 −→ AF = (AB + AC) 3 3

E A

F B

Ç0/= a= ðÍwÍñÍ ÊZ 6, 83*2:-7?,01 ABC = 6M80 8-)12 O 8612J\ )8- −OA+ → −−→ OB + −→ OC = 0 = àxÿxXWxZ ;= 30/2,-1 `= ;**; C −−→ −→ −→ \ −→ −→ −→ H-~8-K OB = OA + AB OC = OA + AC = → −−→ −→ −→ −→ −→ ;2 −OA + OB + OC = 3OA + AB + AC 0\ → −→ −→ /7*L-b68*7?,-\ −AO = (AB + AC) = 610; -+36[-;\ 2/7-b09; [6L6)0 2L-b7*8b-39*8 O -L,6 8-)16 · *,83 89|*/80 83*2:-7?,0K A B 16 ABC = Ç0/= `= ðÍwÍñÍ Z 6 b*18-36N −OA → \ −−→ 0 −→ .-/83-*, .63677*7*.0.*L -K = OB OC ) ) 16[68?\ 8- L06:-,67? OD ~8-:- .63677*7*.0.*L6 .3-N-L08 *3*[ *,83 89|*/80 E 83*2:-7?,016 ABC = àxÿxXWxZ ;= 30/2,-1 ^= H2/8? F C K · )*8b*3869 b*3I0,6 .63677*7*.0.* L6\ 7*|6}69 b .7-/1-/80 OAB = ;*K −→ −→ −→ −−→ −→ *; −OD D E = OA + AF + F D = OA + 1 3

•

•

•

•

•

O

• •

•

B

−−→ −→ −−→ −→ −→ −→ OB + OC, OE = OA + AE = OA + −→ −→ 1 −→ −→ 1 −→ ( AC + AB) = OA + 3 (OC − OA + 3 −−→ −→ − −→ −→ 1 −→ OB − OA) = 3 (OA + OB + OC). −−→ −−→ OE||OD

6K

F 10; -+36[-;\ = Ç0/= ^= ðÍwÍñÍ Z -16[68?\ )8- /8-3-, AB 0 DC )*83*N2:-7?,016 ABCD .63677*7?, 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 -83*[-1 M N \ /-*L0,9J}0M /*K 3*L0, 0N /8-3-,\ .3-N-L08 )*3*[ 8-)12 O .*3*/*)*,09 L06:-,67*M= àxÿxXWxZ ;= 30/2,-1 = Ç6//;-830; +6[0/ ,6 .7-/1-/80\ /-/8-9}0M ïï

A

•

•

D

N

−→ →0 −−→ H 0[ b*18-3-b e = −OA e = OB = 2/8? OC = H −−→ λe 0 OD = µe = - -8,-I*,0J 1 bb*L*,,-;2 −→ \ −−→ \ −→ 0 −−→ 0;*J8 +6[0/2 b*18-3 −OM ON AB DC /7*L2J}0* 1--3L0,68 1

C

1

O

2

M

−−→ 1 1 1 1 , ON = ; λ; µ , 2 2 2 2 −→ −−→ AB = {−1; 1}, DC = {λ; −µ}. −−→ −−→ −→ −−→ OM ||ON ⇔ λ = µ ⇔ AB||DC

−−→ OM =

A

2

B

8/JL6 = Ç0/= = àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*100 Æ\ Ú\ òÅ âä \ :76b6 @\ åÚ Å :76b6 ù@\ åå]\ Æ\ Ú = ðÍwÍñW W S³TÍXxXW´á â]ä \ ò`\ _^\ _]\ _Ú\ ã\ ã]\ aÚ\ a_\ `\ `]\ `ÆÅ âÚä \ 8*;6 = pkkje opogej fhhiçpklg kl mnhofhogp p q mihogilk ogqe' ÊZr

WXXUÌ Tx³xTZ Ï³TxwxxXW´Z ¡ ( ¬ ¤¥ ÉÉ ¡) ©¤º ¡ ¢£ ¬ {O; e , e , e } ¥ £«¦ ¢ ¤ ª O ¬¢ {e } © ¡ V ¬ Ð ©º ¤© ¡§ ªº ¤ A ¡ {O; e } ( º →§ ¤ O ¢£ r = −OA ª ¤ A ¡ {O; e } ¢À£ º ª {x } = {x ; x ; x } ªº r ¬¢ {e } © ¡ V § 610; -+36[-;\ 1

2

3

3

i

i

A

i

i A

i

1 A

2 A

3 A

A

3

rA = xiA ei = x1A e1 + x2A e2 + x3A e3 .

-870)68? b 1--3L0,68,-M [6.0/0 8-)10 -8 b*18-3-b\ 1--3L0,68 >8-+ 8-)*1 +2L*; [617J)68? b 132:7* /1-+10 A(x ; x ; x ) = ï 1 A

2 A

3 A

x3

−→ v = AB

•

A• A •

rB

rA

•

O

B

x2

x1

•

O

Ç0/= ]= Ç0/= Æ= → H2/8? −AB = v \ 8-:L6 0;*J8 ;*/8- /7*L2J}0* O-3;27\ /b9[b6J}0* 1--3L0,68 8-)*1 0 b*18-3-b −→ −→ ] r = r + AB, AB = r − r , x = x + v , v = x − x . ðÍµxñÍXWxZ {/* bI*216[6,,* O-3;27 /.36b*L70b 0 ,6 .7-/1-/80\ 8-7?1- b /72)6* .7-/1-/80 0,L*1/ i .30,0;6*8 Lb6 [,6)*,09\ 6 ,* 830 r = x e = x e + x e = 80 |* O-3;27 /.36b*L70b 0 L79 .39;-M\ 36//;6830b6*;-M 161 /6;-/8-98*7?,M :*-;*830)*/10M -+*18 36[;*3,-K /80 \ .30 ~8-; 0,L*1/ i +2L*8 .30,0;68? b/*:- -L,- [,6)*,0* i = 1 = ÊZ± Tu¶zux uzXuÿxXWx zTxõ zuñxy ³T´µuÌ Z Ï³TxwxxXWxZ Ä Ð ¤ A ¥ B C ¥ ¡ªº »«Ð ¡£¦ ( ¡££ » ¬ ¡» ¡¥ ¡ ª £ ¡ ¬) Ð¥ ¤ B 6= C ¥ º ¢£ À« ¤Û −→ AC Æ (ABC) = (AB, C) = −−→ . CB ß ¤ (ABC) ¢À » Ä ¤ C ¥ ª ( ¡¦) ¢ AB = H2/8? 8-)10 A 0 B L6, 0 83*+2*8/9 ,6M80 8-)12 C \ L*79}2J -83*[-1 ) AB b -8,-I*,00 (ABC) = λ = [ ] .-72 6*; λ(r − r ) = (r − r ) \ -812L6 B

A

A

i A i

B

1 A 1

i B

A

i A

i

i

i B

i A

2 A 2

B

rA + λrB rC = 1+λ

=⇒

xiC

ï¸

xiA + λxiB = , 1+λ

C

i = 1, 2, 3.

C

A

(3)

¼/70 8-)16 E 9b79*8/9 /*3*L0,-M -83*[16 AB \ 8- (ABE) = 1 0 rA + rB , rE = 2

xiE

xiA + xiB = , 2

i = 1, 2, 3.

ÊZÊ

WXXux ³Tu¶zTÍX¶ztu O-3;2703-b6,,-* bI* .-,980* b*18-3,-:- .3-/836,/8b6 .-[b-79*8 / .-;-}?J .3-/8-:- /.0/16 61/0-; .-7,-/8?J -.0/68? /b-M/8b6 /b-+-L,N b*18-3-b :*-;*830)*/1-:- .3-/836,/8b6= 7*L2J}** ,0|* -.3*L*7*,0* 6OK O0,,-:- .3-/836,/8b6 .-[b-79*8 61/0-;680)*/10 [6L68? /6;- :*-;*830)*K /1-* .3-/836,/8b- 161 ;,-|*/8b- 8-)*1= Ï³TxwxxXWxZ ÉÉ ¡ ¢£ ¦ (A, V, ψ) ¥ £«£ ¢ © » A ¥ Ã © ¢Àº £ ¤¥ © ¡ V ¬»£ ψ : A×A → V ¥ £«© ¡£ª¤¦ ¡ ¤ {A, B} » A ¦ → © ¤ £À£ ªÀ« ¢ V ¥ ¬¢¤¦ −AB ¥ ¥ ¡ ª ª 1 ö£ À¬Ð A ∈ A v ∈ V « £ ¤ B → £¥ ¤ −AB = v§ −→ −−→ −→ ª 2 AB+ BC = AC £ À¬Ð A, B, C ∈ A ( ©) = 5OO0,,-* .3-/836,/8b- -+),- -+-[,6)6*8/9 -L,0; /0;b-7-; A = {*1K 8-3,-* .3-/836,/8b- V ,6[b6*8/9 Ñ / 6OO0,,; .3-K /836,/8b-; A = ¢À dim A 6OO0,,-:- .3-/836,/8b6 A ,6[b6*8K /9 36[;*3,-/8? 6//-003-b6,,-:- b*18-3,-:- .3-/836,/8b6 V = 5OO0,,-* .3-/836,/8b- 36[;*3,-/80 n +2L*; -+-[,6)68? /7*L2J}0; -+36[-; A = [ -.3*L*7*,09 6OO0,,-:- .3-/836,/8b6 7*:1- bb-L98/9 /7*L2J}0* Lb6 /--8,-I*,09 → −→ −→ −AA = 0 Å ] AB = −BA = 79 0N L-16[68*7?/8b6 L-/868-),- [6.0/68? 36b*,/8b- 83*2:-7?,016 2 /,6)676 .30 B = C = A \ 6 [68*; .30 C = A = 16|L; b*18-3,; .3-/836,/8b-; ;-|,- 6//-003-b68? 6OO0,,-* .3-/836,/8b- (V, V, ψ) \ .-76:69 ψ : {a, b} 7→ b − a = H-76:69 [68*; V = ) R \.-72 0; 4/86,L638,-*< 6OO0,,-* .3-/836,/8b- R 36[;*3,-/80 n L79 16|L-:- n = ï¾ ◦

◦

n

◦

n

n

¡ b 6OO0,,-; .3-/836,/8b* An 161 0 b :*-;*830)*/1-; .3-K

/836,/8b* ,6[b6*8/9 ,6+-3 {O, e } \ /-/8-9}0M 0[ 8-)10 O ∈ A 0 +6K [0/6 {e } \ i = 1, . . . , n \ 6//-003-b6,,-:- b*18-3,-:- .3-/836,/8b6 V = ªº 8-)10 A -8,-/08*7?,- 3*.*36 {O, e } ,6[b6*8/9 b*18-3 ) ) −→ r = OA = 61 |*\ 161 0 b /72 6* :*-;*830 */1-:- .3-/836,/8b6\ 0;*J8 ;*/8- O-3;27 ]= Ï³TxwxxXWxZ (A, V, ψ) (A , V , ψ ) ª ÉÉÐ ¡º § è¢ª¢¤ ¬» Φ : A → A ¢£ ¢ º É¢ ÉÉÐ ¡¥ Φ ªÑ ¢É¢ Ñº Ð Ð ¡¥ « ¢ º É¢ ϕ : V → V ¦¥ ¤ −−→ −→ Ú A = Φ(A), B = Φ(B). ϕ(AB) = A B Þ « ¢É¢ Φ : A → A ¥ ÉÉ ¡ A A ¢À£ ¢É§ ¼/70 Φ : A → A · 0[-;-3O0[; 6OO0,,N .3-/836,/8b\8- 0[-;-3O0[; b*18-3,N .3-/836,/8b ϕ : V → V \ -.3*L*79*;M /--8,-I*,0*; Ú\ ,6[b6*8/9 0[-;-3O0[;-;\ Ñ / 0[-;-3O0[;-; Φ = 7*L2*8 -8;*808?\ )8- 0[ 2/7-b09 Ú .3*|L* b/*:- /7*L2*8\ )8- */70 A = Φ(A) \ B = Φ(B) \ C = Φ(C) \ D = Φ(D) \ 8−−→ −−→ −→ −−→ ò AB = CD =⇒ A B = C D . -7?1- .30 b.-7,*,00 2/7-b09 ò -8-+36|*,0* Φ : A → A 0,L2032*8 ,*1-8-3-* -8-+36|*,0* ϕ : V → V = TxwuxXWxZ ôÀ¬ ª ÉÉ ¡ ª¦ ¢ A A ¢É§ éuyÍêÍzxë¶ztuZ H2/8? {O, e } · .3-0[b-7?,M 3*.*3 b A \ 6 {O , e } · .3-0[b-7?,M 3*.*3 b A \ 8-:L6 -8-+36|*,0* Φ \ -8,-/9}** 8-))1* M ∈ A \ 0;*J}*M 1--3L0,68 (x ) -8,-/08*7?,- 3*.*36 {O, e } \ 8- 12 M ∈ A \ 0;*J}2J 8* |* /6;* 1--3L0,68 (x ) -8,-/08*7?,- 3*.*36 {O , e } \ 9b79*8/9 0[-;-3O0[;-; Φ : A → A = [ ~8-:- .3*L7-|*,09 /7*L2*8\ )8-\ / 8-),-/8?J L- 0[-;-3O0[;6\ /2K }*/8b2*8 8-7?1- -L,- 6OO0,,-* .3-/836,/8b- L6,,-M 36[;*3,-/80 n = { ïÁ i

n

i

n

i

A

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

n

0 n

i

0

n

0

0 n

n 0 n

i M

i M

n

0

i

0 n

0

0

i

i

)6/8,-/80\ b/* 6OO0,,* .3-/836,/8b6 36[;*3,-/80 3 0[-;-3O, :*-;*8K 30)*/1-;2 .3-/836,/8b2 0 6OO0,,-;2 .3-/836,/8b2 R = ðÍµxñÍXWxZ H-/3*L/8b-; bb*L*,09 6OO0,,N .3-/836,/8b ; .-72)6*; b-[;-|,-/8? ,* 8-7?1- /83-:- -.0/68? :*-;*830)*/1-* 83*N;*3,-* .3-/8K 36,/8b- 0 :*-;*830)*/12J .7-/1-/8?\ ,- 0 36//;6830b68? 0N ;,-:-;*3,* 6,67-:0= 8;*80;\ )8- -/86*8/9 *}* -.3*L*708? 8610* :*-;*830)*/10* .-K ,9809 161 36//8-9,0* ;*|L2 8-)16;0 0 2:-7 ;*|L2 .39;;0= 8- +2L*8 -/2}*/8b7*,- ,0|* / .-;-}?J -.*3600 /16793,-:- .3-0[b*L*,09 b*18-K 3-b= Ï³TxwxxXWxZ W ⊂ V ¡ª¡ ¡º V ¥ Ñ n º ÉÉ ¡ A § À ¢ m (m º¡À) A (0 6 m 6 n) ¥ ¡£À« ¡ª¡º ¡Ðª£«¦ ¤¢ ¤ A À«¦ W ¥ ¢£ ªÀ« ¡ª» A 3

m

n

n

n

n

m

n

−−→ πm = {M ∈ An | AM ∈ Wm }.

{ :*-;*830)*/1-; .3-/836,/8b* K.7-/1-/80 · ~8- .39;*\6 ]K.7-/1-/80 -+),* .7-/1-/80= · ~8[ ~8-:- -.3*L*7*,09 /7*L2*8\ )8- 83-M16 (π , W , ψ| ) \ :L* ψ| · -:36K ,0)*,0* -8-+36|*,09 ψ ,6 π × π \ · mK;*3,-* 6OO0,,-* .3-/836,/8b- = 61-* .3-/836,/8b- ,6[b6*8/9 ¡ª¡ b A = ÊZ TWµxTUZ 7*L2J}0* .30;*3 .-16[b6J8\ 161 6OO0,,* 1--3L0,68 0 .3-/8-* -8,-I*,0* 83*N 8-)*1 ;-:28 0/.-7?[-b68?/9 .30 3*I*,00 :*-;*830)*/10N [6L6)= ðÍwÍñÍ Z *-3*;6 °*,*769= 6 /8-3-,6N AB \ BC 0 CA 83*2:-7?,016 ) ) ABC L6, 8- 10 C \ A 0 B 8610*\ 8- (BCA ) = λ \ (CAB ) = λ 0 ) ) (ABC ) = λ = -16[68?\ 8- 8- 10 A \ B 0 C 7*|68 ,6 -L,-M .39;-M 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 λ λ λ = −1 = àxÿxXWxZ ;= 30/2,-1 Ú = Ç6//;-830; 6OO0,,M 3*.*3 {O, e } \ :L* −→ −→ { O = A \ e = AB \ e = AC = ~8-; 3*.*3* b*3I0, 83*2:-7?,016 0 36/K ïÈ m

m

m

π

π

m

n

0

0

0

0

3

0

0

0

1

0

2

0

1 2 3

i

1

2

B0 e2

C A0

O=A

C0

e1

/;6830b6*;* ,6 *:- /8-3-,6N 8-)10 0;*K J8 /7*L2J}0* 1--3L0,68 A(0; 0) \ B(1; 0) \ C(0; 1) \ A0

λ1 1 1+λ1 ; 1+λ1

, B0 0;

1 1+λ2

, C0

λ3 1+λ3 ; 0

H3-/8* b)0/7*,09 .-16[b6J8\ )8-

B

.

⇐⇒ λ λ λ = −1. Ç0/= Ú = ðÍwÍñÍ Z { .3-/836,/8b* [6L6, )*83* 8-)10 A \ B \ C \ D = -)16 E · /*3*L0,6 AB \ 6 8-)16 F · /*3*L0,6 CD = -16[68?\ )8−−−→ −−−→ C0 A0 ||C0 B0

1 2 3

−→ 1 −→ −−→ EF = (AC + BD). 2

àxÿxXWxZ ;= 30/2,-1 ò=

C

•

F •

−→ (rC + rD ) − (rA + rB ) EF = rF − rE = = 2 −→ −−→ (rC − rA ) + (rD − rB ) AC + BD = = . 2 2

• •

D

B

•

A•

E

Ç0/= ò= ðÍwÍñÍ Z 6M80 36L02/Kb*18-3 8-)10 E · *,836 89|*/80 83*2:-7?,016 ABC \ */70 0[b*/8, 36L02/Kb*18-3 *:- b*3I0, = àxÿxXWxZ ;= 30/2,-1 _= H2/8? F · /*3*L0,6 C BC \ 8-:L6 (BCF ) = 1 0 (AF E) = 2 = 8/JL6 F E •

•

rB + rC , 2 rA + 2rF rA + rB + rC rE = = . 3 3 rF =

• •

A

•

Ç0/= _=

B

ðÍwÍñÍ Z -16[68?\ )8- .39;*\ /-*L0,9J}0* /*3*L0, .3-80b-.-7-|K ,N 3*+*3 8*836~L36\ .*3*/*16J8/9 b -L,-M 8-)1* 0 L*798/9 b ,*M .-.-76;= -16[68?\ )8- b ~8-M |* 8-)1* .*3*/*16J8/9 0 .39;*\ /-*L0,9J}0* b*3K I0, 8*836~L36 / *,836;0 89|*/80 .3-80b-.-7-|,N :36,*M= ïÔ

D

àxÿxXWxZ ;= 30/2,-1 ã= +-[,6)0; b*3K I0, 8*836~L36 +21b6;0 A \ B \ C \ D = H2/8? E · /*3*L0,6 AB \ F · /*3*L0,6 CD \ 6 G · /*3*L0,6 EF = ;**;

•

F •

C G •

•

A

E

• •

rE =

•

B

rA + rB , 2

rF =

rC + rD , 2

r +r r +r +r +r r = = . Ç0/= ã= 2 4 -)16 )G · )*,83 89|*/80 8*836~L36= )*b0L,-\ )8- ~8- 0 */8? 216[6,,69 b [6L6 * 8- 16= àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*109 òÅ âä \ :76b6 @\ åÚ Å :76b6 ù@@\ åå\ ]\ Æ= ðÍwÍñW W S³TÍXxXW´á â]ä \ ãò\ ã_\ a^\ a\ _a\ _`\ ã^\ ãÆ\ ãò\ ã`\ a^\ ò]\ òÚ\ ]ãÚ\ ]ã_Å âÚä \ 8*; ]ò= E

F

A

B

C

D

G

flnikhe mihpqeçekpe qefghihq '

Zr TuxyÎWW txyzuTut XÍ ³T´µS W ³u¶yu¶zëZ

Ç6//;-830; b :*-;*830)*/1-; .3-/836,K /8b* .7-/1-/8? π 0 .39;2J ` \ .*3*/*16K A B J}2J π b -L,-M 8-)1* = +-[,6)0; /0;K b-76;0 V (`) 0 V (π) .-L.3-/836,/8b6 b V \ /-/8-9}0*\ /--8b*8/8b*,,-\ 0[ b*18-K C 3-b .63677*7?,N .39;-M ` 0 .7-/1-/80 O π H π = 3-*180329 b*18-3 ,6 ` .63677*7?,) ` π 0 ,6 π .63677*7?,- ` \ .-72 0; -8-+36K |*,09 pr : V → V (`) 0 pr : V → Ç0/= a= V (π) = TxwuxXWxZ pr : V → V (`) pr : V → V (π) ¦ ¬º »£= éuyÍêÍzxë¶ztuZ {+*3*; b V +6[0/ {e , e , e } 861-M\ )8- e ∈ V (`) \ 6 {e , e } · +6[0/ b V (π) = -:L6 L79 pr 0;**; pr (λv + µw) = pr (λv e + ïÙ 1

2

3

3

`

1

3

π

2

`

3

1

3

2

3

2

3

π

1

π

2

2

3

π

1

1

π

i

i

µwi ei ) = prπ (λv i +µwi )ei = (λv 2 +µw2 )e2 +(λv 3 +µw3 )e3 = λ(v 2 e2 +v 3 e3 )+ µ(w2 e2 + w3 e3 ) = λ prπ (v) + µ prπ (w)

= 5,67-:0),- L-16[b6*8/9 70,*M,-/8? -8-+36|*,09 pr = Ï³TxwxxXWxZ ¿À ©¤ ¡ ¢À º À ¡£À¥ ¡£À¥ ¦ ¢É ª ¢ ªÐ ¢»Ð ¡¦§ Ò¡£À« ¢£ £¦ ¦ ¥ ¡ © ¡ª ¡º § `

a

`

Ç0/= `= 79 -+-[,6)*,09 -/0 +2L*; 0/.-7?[-b68? /0;b-7 ` ↑ = ðÍµxñÍXW´Z = ,82080b,- 9/,-* .-,980* ,6.36b7*,09\ 0/.-7?[2*;-* b -.3*L*7*,00 -/0\ ;-|,- /7*L2J}0; -+36[-; O-3;670[-b68? L79 /72K )69 .3-0[b-7?,-:- b*18-3,-:- .3-/836,/8b6 V 6\ [,6)08\ 0 L79 7J+-K :- 6OO0,,-:- .3-/836,/8b6= 6 ;,-|*/8b* V \ {0} ,*,27*bN b*18-K 3-b .3-/836,/8b6 V bb*L*; -8,-I*,0* ~1b0b67*,8,-/80\ .-76:69 a ∼ b \ */70 a = λb 0 λ > 0 = -:L6 ¡ b V · ~8- ~7*;*,8 O618-3K ;,-|*/8b6 V \ {0}/∼ = { .3-/836,/8b* V /b-+-L,N b*18-3-b :*-;*8K 30)*/1-:- .3-/836,/8b6 16|LM 861-M 176// ~1b0b67*,8,-/80 /-L*3|08 3-b,- -L0, ª¤¦ 861 ,6[b6J8 b*18-3 L70, *L0,06= H-~8-;2 ;,-|*/8b- ,6.36b7*,0M b :*-;*830)*/1-; .3-/836,/8b* ,6N-L08K /9 b- b[60;,--L,-[,6),-; /--8b*8/8b00 /- ;,-|*/8b-; *L0,0),N b*18-K 3-b= ¼/70 -87-|08? b/* *L0,0),* b*18-3 -8 -L,-M 8-)10\ 8- 0N 1-, -.0I28 /O*32 *L0,0),-:- 36L02/6= H-~8-;2 ;,-|*/8b- ,6.36b7*,0M b :*-K ;*830)*/1-; .3-/836,/8b* ,6N-L08/9 b- b[60;,--L,-[,6),-; /--8b*8/8b00 /- ;,-|*/8b-; 8-)*1 /O*3 *L0,0),-:- 36L02/6= 5,67-:0),; -+36[-;\ ;,-|*/8b- ,6.36b7*,0M ,6 :*-;*830)*/1-M .7-/1-/80 ,6N-L08/9 b- b[60;K ,--L,-[,6),-; /--8b*8/8b00 /- ;,-|*/8b-; 8-)*1 -132|,-/80 *L0,0),-:36L02/6= ]= { /72)6* ,*-30*,803-b6,,-M .39;-M ` .39;-M 70,00 b -+),-; n

n

n

n

3

n

Ý

/;/7*\ ** ,6.36b7*,0* [6L6*8/9 7J+; ,*,27*b; b*18-3-;\ .63677*7?K ,; ~8-M .39;-M= { ~8-; /72)6* 8*3;0, ¡ b V -[,6)6*8 ~7*K ;*,8 O618-3K;,-|*/8b6 V \ {0}/∼ .- -8,-I*,0J ~1b0b67*,8,-/80\ -.3*K L*7*,,-;2 /7*L2J}0; -+36[-; a ∼ b \ */70 a = λb \ :L* λ ∈ R ;-|*8 +8? 161 .-7-|08*7?,;\ 861 0 -83068*7?,;= °,-|*/8b- ,6.36b7*,0M b ~8-; /;/7* b :*-;*830)*/1-; .3-/836,/8b* ,6N-L08/9 b- b[60;,--L,-K [,6),-; /--8b*8/8b00 /- ;,-|*/8b-; .63 L06;*8367?,- .3-80b-.-7-|,N 8-)*1 /O*3 *L0,0),-:- 36L02/6= ( ~8-:- ;,-|*/8b6 */8? /b-* ,6[b6,0* .3-*180b,69 .7-/1-/8?= Æ= P61 ~7*;*,8 O618-3K;,-|*/8b6 V \ {0}/∼ \ ,6.36b7*,0* [6L6*8/9 7J+; /b-0; .3*L/86b08*7*; · ,*,27*b; b*18-3-;= H-~8-;2 b L67?,*MK I*;\ :-b-39 - ,*1-8-3-; ,6.36b7*,00\ +2L*; -+-[,6)68? *:- 7J+; .3*LK /86b79J}0; *:- b*18-3-;= Ï³TxwxxXWxZ ` ↑ ¡£À« a ¥ b ∈ V ¡¢¦ § ¿©¦ ¡Ñ¦ b ` ↑ ¢£ ¤ pr (b) = |b| cos ϕ \ :L* ϕ · 2:-7 ;*|L2 a 0 b = n

n

n

3

`↑

b ϕ a ^ 0/ ] Ç = = TxwuxXWxZ ¿¬» pr : V 3 b 7→ pr ¬»= éuyÍêÍzxë¶ztuZ Ç6//;-830; b V +6K [0/ {e , e , e } \ :L* e = \ 6 e 0 e · ,*1-770,*63,* b*18-3 -38-:-,67?,* b*18-32 e = -:L6 pr : b 7→ b \ /7*L-K b68*7?,-\ pr · 70,*M,-* -8-+36|*,0* = 3

`↑

`↑ (b)

∈R

¦

3

1

2

3

1

1

`↑

a |a|

2

1

3

e2

e3

•

e1

`↑

Ç0/= ]= H-/1-7?12 .3-*100 ,6 b/* -/0 ` ↑ / O01/03-b6,,; ,6.36b79J}0; b*18-3-; a · ~8- -L,- 0 8- |* -8-+36|*,0*\ 8- L79 -8-+36|*,09 pr Ü

`↑

+2L*; 0/.-7?[-b68? 861|* -+-[,6)*,0* pr = Z± syÍ´TXux ³TuWêtxwxXWxZ Ï³TxwxxXWxZ a ¥ b ¢ V ¥ ϕ © »ª § ú£ ¡¢ª a b ¢£ ¤ a

3

(a, b) = |a||b| cos ϕ.

xuµxzTWñx¶yWx ¶tuÌ¶ztÍ ¶yÍ´TXuVu ³TuWêtxwxXW´Z

7*L2J}0* /b-M/8b6 /16793,-:- .3-0[b*L*,09 b*18-3-b b8*16J8 ,*.-K /3*L/8b*,,- 0[ -.3*L*7*,09 a ⊥ b ⇐⇒ (a, b) = 0 .- -.3*L*7*,0J /)086*8/9\ )8- 0 ⊥ b L79 7J+-:- b ∈ V = ] H3*L.-76:69\ )8- 2:-7 ϕ ;*|L2 a 0 b 2L-b7*8b-39*8 /--8,-I*,0J 0 6 ϕ 6 π \ 0;**; (a, b) > 0 ⇐⇒ ϕ < \ (a, b) < 0 ⇐⇒ ϕ > = Æ (a \ a) = |a| = Ú pr (b) = = VxvTÍWñx¶yWx ¶tuÌ¶ztÍ ¶yÍ´TXuVu ³TuWêtxwxXW´Z 1 = (a, b) = (b, a) = 2 = (λa + µb, c) = λ(a, c) + µ(b, c) = 3 = (c, λa + µb) = λ(c, a) + µ(c, b) = 4 = a 6= 0 =⇒ (a, a) > 0 = éuyÍêÍzxë¶ztuZ 1 0 4 b8*16J8 ,*.-/3*L/8b*,,- 0[ -.3*L*7*,09 = 3 /7*L2*8 0[ 1 0 2 = -16|*; 2 = ¼/70 c = 0 \ 8- 2 b.-7,9*8/9 -)*b0L,; -+36[-;= ¼/70 c 6= 0 \ 8(λa + µb, c) = |c|pr (λa + µb) = λ|c|pr (a) + µ|c|pr (b) = λ(a, c) + µ(b, c) = Ï³TxwxxXWxZ V W ¡§ ¿¬»º ϕ : V × V → W ¢£ ¬¦¥ ϕ(λa + µb, c) = λϕ(a, c) + µϕ(b, c) \ ϕ(c, λa + µb) = λϕ(c, a) + µϕ(c, b) = 3

π 2 π 2

2

(a,b) |a|

a

◦ ◦ ◦ ◦

◦

◦

◦

◦

◦

◦

◦

c

c

ï

c

610; -+36[-;\ ϕ : V × V → W · +070,*M,-\ */70 .30 .3-0[b-7?,-; O01/03-b6,,-; a ∈ V 16|L-* 0[ -8-+36|*,0M V 3 v 7→ ϕ(a, v) ∈ W \ V 3 v 7→ ϕ(v, a) ∈ W \ 9b79*8/9 70,*M,; = sxw¶ztWxZ ú¦ 2 3 ¡ Ã ¬¦º ¬»£ g : V × V 3 {a, b} 7→ (a, b) ∈ R. ZÊ TWµxTUZ 7*L2J}0* .30;*3 .-16[b6J8 161 /16793,-* .3-0[b*L*,0* ;-|*8 .30K ;*,98?/9 .30 3*I*,00 36[70),N :*-;*830)*/10N [6L6)= ðÍwÍñÍ rZ *-3*;6 M7*36= -16K A [68?\ )8- /2;;6 1b6L368-b /8-3-, )*K A 83*N2:-7?,016 A A A A 36b,6 /2;K ;* 1b6L368-b *:- L06:-,67*M 0 2)*8b*K E F 3*,,-:- 1b6L3686 36//8-9,09 ;*|L2 /*K 3*L0,6;0 L06:-,67*M= A A àxÿxXWxZ ;= 30/2,-1 ]]=+-[,6)0; 0/ ]] Ç = = /*3*L0, L06:-,67*M A A 0 A A \ /-K -8b*8/8b*,,-\ +21b6;0 E 0 F \ 0 .2/8? r · 36L02/Kb*18-3 b*3I0, A = { [6L6)* 83*+2*8/9 L-16[68?\ )8- 0;**8 ;*/8- /7*L2J}** 36b*,/8b- ◦

◦

3

•

3

4

•

3

1

•

2

3

4

1

3

•

•

•

•

1

2

2

i

4

i

(r2 − r1 )2 + (r3 − r2 )2 + (r4 − r3 )2 + (r1 − r4 )2 = 2 (r + r ) (r + r ) 2 4 1 3 − = (r4 − r2 )2 + (r3 − r1 )2 + 4 . 2 2

79 L-16[68*7?/8b6 ~8-:- 36b*,/8b6 L-/868-),- 36/138? /1-+10 0 /-+368? .-L-+,* )7*,= ðÍwÍñÍ rrZ -)16 M 36/.-7-|*,6 b,2830 b.217-:- nK2:-7?,016 P = ) A A . . . A = -16[68?\ 8- ,6ML*8/9 86169 /8-3-,6 A A ~8-:- nK2:-7?,016\ )8- -/,-b6,0* .*3.*,L0127936\-.2}*,,-:- 0[ 8-)10 M ,6 A A 9b79*8/9 b,283*,,*M 8-)1-M -83*[16 A A = àxÿxXWxZ ;= 30/2,-1 ]Æ= 79 2L-+/8b6 -+-[,6)*,0M +2L*; /)0868?\ )80,L*1/ i = 1, 2, . . . , n .3*L/86b79*8 /-+-M -/868-1 -8 L*7*,09 ,6 n \ 8- */8?\ n ∼ 0 \ n + 1 ∼ 1 0 861 L67** = 61 161 ;,-:-2:-7?,01 P · b.217M\ 1

2

n

i

i+1

i

i

i+1

i+1

8- -, 9b79*8/9 -+*L0,*,0*; ,*.*3*/*16J}0N/9 83*2:-7?,01-b M A A \ ) i = 1, 2, . . . , n = 3*+2*8/9 L-16[68?\ 8- ,6ML*8/9 86169 /8-3-,6 A A \ )8- b 83*2:-7?,01* M A A 2:7 ∠M A A 0 ∠M A A -+6 9b79J8K /9 -/83;0= H3*L.-7-|0;\ )8- /2}*/8b2*8 861-M ;,-:-2:-7?,01\ )8- b 16|L-; 83*2:-7?,01* M A A \ i = 1, 2, . . . , n -L0, 0[ 2:7-b ∠M A A 070 ∠M A A +-7?I* 070 36b*, = H2/8? .30 ~8-;\ L79 -.3*L*7*,,-/80\ ∠M A A > \ 8-:L6 ∠M A A < \ A A .-~8-;2 ∠M A A > \-812L6 /7*L2*8\ O )8- ∠M A A > \ i = 1, 2, . . . , n = {K +*3*; 8-)12 M [6 ,6)67- 3*.*36 M = A −−→ A O 0 .2/8? r = M A · 36L02/Kb*18-3 b*3I0, A = ;**; (r , r − r ) 6 P (r , r − r ) 6 0 0 = - 8-:L6 2 A A P P ⇐⇒ 2 r −2 (r , r ) 6 0 Ç0/= ]Æ= P ⇐⇒ (r − r ) 6 0 = Z syÍ´TXux ³TuWêtxwxXWx t yuuTwWXÍzÍõZ H2/8? b .3-/836,/8b* V 070 V \ */70 36//;6830b6J8/9 b*18-3 ,6 :*-K ;*830)*/1-M .7-/1-/80 b+36, ,*1-8-3M +6[0/ {e } = {e , e , e } b /72K )6* .7-/1-/80 {e } = {e , e }= {)0/70; /16793,-* .3-0[b*L*,0* b*18-K 3-b a = a e 0 b = b e = ;**; _ (a, b) = (a e , b e ) = a b (e , e ) = g a b , :L* ) ã g = (e , e ), i, j = 1, 2, 3 (i, j = 1, 2 b /72 6* V ). [ O-3;27 _ /7*L2*8\ )8- L79 b)0/7*,09 /16793,-:- .3-0[b*L*,09 b*18-3-b\ [6L6,,N /b-0;0 1--3L0,686;0 b +6[0/* {e } \ ,*-+N-L0;- /,6)6K 76 b)0/708? ;68302 k

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1

1

i

1

2

ij

11

12

21

22

C

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0 n

i M

-8,-/08*7?,- 3*.*36 {O , e } \ 9b79*8/9 0[-;-3O0[;-; *b170L-bN 6OO0,,N .3-/836,/8b Φ : E → E = [ ~8-:- .3*L7-|*,09 /7*L2*8\ )8-\ / 8-),-/8?J L- 0[-;-3O0[;6\ /2}*K /8b2*8 8-7?1- -L,- *b170L-b- 6OO0,,-* .3-/836,/8b- L6,,-M 36[;*3,-K /80 n = { )6/8,-/80\ b/* *b170L-b 6OO0,,* .3-/836,/8b6 36[;*3,-/80 3 0[-;-3O, :*-;*830)*/1-;2 .3-/836,/8b2 0 *b170L-b2 6OO0,,-;2 .3-K /836,/8b2 R = )*b0L,-\ b/9169 mK.7-/1-/8? π b E 9b79*8/9 mK;*3,; *b170L-b; 6OO0,,; .3-/836,/8b-;= ðÍµxñÍXWxZ 6 b/91-; b*18-3,-; .3-/836,/8b* V \ [6L6b .3-0[b-7?,; -+36[-; /0;;*830),2J .-7-|08*7?,- -.3*L*7*,,2J +070,*M,2J O-3;2 g : V ×V → R \ ;-|,- bb*/80 /83218232 *b170L-b6 b*18-3,-:- .3-/836,K /8b6= 8- ;-|,- -/2}*/8b08?\ ,6.30;*3\ /7*L2J}0; -+36[-; b+36b .3-K 0[b-7?,M +6[0/ e b V \ .-7-|08? g(a, b) = δ a b = a b +a b +. . .+a b = (16[6,,; /.-/-+-; ;-|,- bb*/80 861|* ,-b-* /16793,-* .3-0[b*L*,0* b *b170L-b-; b*18-3,-; .3-/836,/8b* b 1-8-3-; -L,- /16793,-* .3-0[b*L*K ,0* 2|* */8?= ¼/70 bb*/80 /83218232 *b170L-b6 .3-/836,/8b6 ,6 b*18-3K ,-; .3-/836,/8b* V \ 6//-003-b6,,-; / 6OO0,,; .3-/836,/8b-; A \ 8- 8*; /6;; ,6 .3-/836,/8b* A +2L*8 bb*L*,6 /83218236 *b170L-b6 6OK O0,,-:- .3-/836,/8b6= {b*L*,0* /8321823 *b170L-b6 .3-/836,/8b6 b 6OO0,,-; .3-/836,/8K b* ;-|*8 +8? 0/.-7?[-b6,- .30 3*I*,00 1-,13*8,N [6L6)= H30;*3-; ;-|*8 /72|08? /7*L2J}69 [6L6)6= ðÍwÍñÍ rZ 6 6OO0,,-M .7-/1-/80 A L6,6 836.*09 ABCD / .6367K 7*7?,;0 /8-3-,6;0 AB 0 CD = -16[68?\ )8- 8-)16 .*3*/*)*,09 L06:-K ,67*M AC 0 BD 7*|08 ,6 .39;-M\ .3-N-L9}*M )*3*[ /*3*L0, -/,-b6,0M AB 0 CD = àxÿxXWxZ H2/8? O · /*3*L0,6 AB \ 6 E · /*3*L0,6 CD /;= 30/= ]ã= {b*K −→ 0 −−→ L*; ,6 A /16793,-* .3-0[b*L*,0*\ .- -8,-I*,0J 1 1-8-3-;2 −OB OE · ) *L0,0 ,* b[60;,- -38-:-,67?,* b*18-3= ,6+|*,,69 ~80; /16793,; .3-0[b*L*,0*;\ A .3*b36}6*8/9 b *b170L-b2 .7-/1-/8? E \ ,6 1-8-3-M 836.*09 ABCD 9b79*8/9 36b,-+-),-M 0\ /7*L-b68*7?,-\ /0;;*830),-M -8,-/08*7?,- .39;-M OE = ¸Ü 0

(xiM )

0

i

0 n

n

3

m

n

n

n

n

i

n

ij

i j

1 1

2 2

n

n

n

2

2

2

n n

2

S0

S

E

D

A

O

D0

C A0

B

E0

C0

O0

B0

Ç0/= ]ã= { *b170L-bN .3-/836,/8b6N bb-L98/9 .-,9809 ;-L279 b*18-36\ 2:76 ;*|L2 b*18-36;0 0 36//8-9,09 ;*|L2 8-)16;0= ª b*18-36 a b E ,6[b6*8/9 )0/7n

|a| =

p (a, a).

© ;*|L2 b*18-36;0 a 0 b b E ,6[b6*8/9 )0/7- ϕ ∈ [0, π] \ -L,-[,6),n

-.3*L*79*;-* /--8,-I*,0*;

cos ϕ =

(a, b) . |a||b|

£ ;*|L2 8-)16;0 A 0 B b En ,6[b6*8/9 )0/7−→ d(A, B) = |AB|.

O-3;2703-b6,,* b ,6/8-9}*; .636:36O* -.3*L*7*,09 .-[b-79J8 61K /0-;680[03-b68? :*-;*830)*/1-* .3-/836,/8b- 0 :*-;*830)*/12J .7-/1-/8?\ -.3*L*799 0N\ /--8b*8/8b*,,-\ 161 83*N;*3,-* 0 Lb2;*3,-* *b170L-b 6OK O0,,* .3-/836,/8b6= { L67?,*MI*; L79 :*-;*830)*/10N .3-/836,/8b6 0 .7-/1-/80 +2L*; 0/.-7?[-b68?\ /--8b*8/8b*,,-\ -+-[,6)*,09 E 0 E = 5//-K 003-b6,,* .3-/836,/8b6 b*18-3-b +2L*; -+-[,6)68? /7*L2J}0; -+36K [-; E 0 E = àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*100 Æ\ Ú Å âä \ :76b6 @\ åÚ\ :76b6 ù@\ å]Å âòä \ :76b6 ]\ åå\ ]= ðÍwÍñW W S³TÍXxXW´á â]ä \ `a\ ``\ ]^\ ]^]\ ]^Æ\ ]^Ú\ ]^a\ ]^\ ]\ ]ò\ ]_\ ]`\ òò\ _Æ\ _ò\ òÚ\ òò\ ò_\ ò`\ _^Å âÚä \ 8*; _a= 3

3

2

¸ï

2

#meilp mhqhihgl qefghil kl hn kl mnhofhogp ' α Zr utuTuz txyzuTÍ XÍ SVu α XÍ uTWxXzWTutÍXXuÌ ³u¶yu¶zWZ

Q*-;*830)*/169 *b170L-b6 .7-/1-/8? E ,6[b6*8/9 ¦\*/K 70 ,6 ,*M [6O01/03-b6,- .-7-|08*7?,-* ,6.36b7*,0* -8/)*86 2:7-b= Â6[0/ ) {e , e } ,6[b6*8/9 ¡\ */70 ,6.36b7*,0* 1368 6MI*:- .-b-3-86 -8 b*18-36 e 1 b*18-32 e 9b79*8/9 .-7-|08*7?,;= Ç*.*3 {O; e , e } ,6[K b6*8/9 ¡\ */70 .36b; 9b79*8/9 +6[0/ {e , e } = Â6[0/ 0 3*.*3\ ,* 9b79J}0*/9 .36b;0\ ,6[b6J8/9 = 6 )*38*|6N .30,98- /)0868? .-7-|08*7?,; ,6.36b7*,0* -8/)*86 2:7-b .3-80b )6/-b-M /83*710= H36bM -38-,-3;03-b6,,M 3*.*3 ,6 .7-/1-/80 .30,98- -+-[,6)68? /7*L2J}0; -+36[-; {O; i, j} = P--3L0,68 b*18-36 a 0 8-)10 A b ~8-; /72K )6* -+-[,6)6J8\ /--8b*8/8b*,,-\ /7*L2J}0; -+36[-; {x ; y } 0 (x ; y ) = ¹-3;276 L79 /16793,-:- .3-0[b*L*,09 .30,0;6*8 b0L (a, b) = x x + y y = {*18-3 e(ϕ) \ .-72)6J}0M/9 .-b-3-8-; b*18-36 i ,6 -30*,803-b6,,M 2:-7 ϕ \ 0;**8 /7*L2J}0M b0L 2

1

2

1

2

1

1

2

2

a

a

A

A

a b

a b

e(ϕ) = i cos ϕ + j sin ϕ.

/.-7?[29 ~8-8 b*18-3\ .3-0[b-7?,M b*18-3 a ;-|,- .3*L/86b08? b b0L*

y a

a = |a|e(ϕa ),

:L* ϕ · 2:-7\ ,6 1-8-3M ,2|,- .-b*3,28? b*18-3 i \ )8-+ *:- ,6.36b7*,0* /-b.67- / ,6K .36b7*,0*; b*18-36 a = H30 ~8-; a

xa = |a| cos ϕa ,

O

Ç0/= ]a=

ya = |a| sin ϕa .

Ç6//;-830; -8-+36|*,0*

Φα : E2 3 a = |a|e(ϕa )

ϕa

7−→

Φα (a) = |a|e(ϕa + α) ∈ E2 ,

.-b-36)0b6J}** b*18-3 .7-/1-/80 ,6 2:-7 α = ¸

x

;**;

y

Φα (a) = |a| i cos(ϕa + α) + j sin(ϕa + α) =

Φα (a) α

= i(xa cos α − ya sin α) + j(ya cos α + xa sin α).

610; -+36[-;\ b 1--3L0,686N -8-+36|*,0* Φ 0;**8 b0L

a ϕa x

O

α

Ç0/= ]`=

]Æ TxwuxXWxZ Φ : E → E ¢É¢ ªÐ Ð ¡= L79 *:- L-K 8- 28b*3|L*,0* -)*b0L,- 0[ :*-;*830)*/10N /--+36|*,0M\ 16[68*7?/8b6 ;-|,- 861|* b-/.-7?[-b68?/9 O-3;27-M ]Æ= ¹-3;272 ]Æ 2L-+,- [6.0/b68? b /7*L2J}*; ;6830),-; b0L* Φα : {xa ; ya } 7→ {xa cos α − ya sin α; xa sin α + ya cos α}. 2

α

Φα :

xa ya

2

!

cos α − sin α sin α cos α

7−→

{ 70,*M,-M 67:*+3* ;68306

cos α − sin α sin α cos α

!

xa ya

!

]Ú

!

,6[b6*8/9 ;6830*M 70,*M,-:- -.*368-36 Φ b L6,,-; +6[0/* /;= âä \ :76b6 ù@@@\ å]= ¼* /8-7+ /-/8-98 0[ 1--3L0,68 b*18-3-b Φ (i) 0 Φ (j) = ðÍµxñÍXWxZ H2/8? Φ : V → V · .3-0[b-7?,-* 70,*M,-* -8-+36|*,0* 0 {e } \ i = 1, . . . , n \ · ,*1-8-3M +6[0/ b V = -:L6 α

α

n

α

n

i

n

Φ(a) = Φ(ai ei ) = ai Φ(ei ) = Φki ai ek ,

610; -+36[-;\ */70 b = Φ(a) \ 8-

Φ(ei ) = Φki ek .

]ò ¯6.0/b69 1--3L0,68 {a } 0 {b } b*18-3-b a 0 b /8-7+6;0 0 36//;6830K b69 ;68302   bk = Φki ai .

i

i

Φ11 Φ12 . . . Φ1n   2 2  2  . . . Φ Φ Φ n  2 1 (Φki ) =  ,    n n n Φ1 Φ2 . . . Φ n

== == = = = == ¸¸

]_

/8-7+ 1-8-3-M /-/8-98 0[ 1--3L0,68 -+36[-b Φ(e ) b*18-3-b +6[0/6 {e } -8,-/08*7?,- ~8-:- |* +6[0/6\ /--8,-I*,0* ]ò ;-|,- [6.0/68? b ;6830)K ,-; b0L*      i

i

a1 Φ1 Φ12 . . . Φ1n b1   2   12   b   Φ1 Φ22 . . . Φ2n   a2   =         n n n n b Φ1 Φ2 . . . Φ n an

==

== == = = = ==

==

]ã

    

°68306 ]_ ,6[b6*8/9 Ñ¦ ¦© ¬»£ ì¡î Φ b +6[0/* {e } = ðÍwÍñÍ r Z {b*/80 O-3;272 ]Ú 161 )6/8,M /72)6M -+}*M O-3;27 ]ã= 79 -.*3600 .-b-3-86 b*18-36 ,6 2:-7 .30,98- 0/.-7?[-b68? /7*L2K J}** -+-[,6)*,0* i

π 2

Φ π2 (a) = [a].

¹-3;276 ]Æ .30 α = .30,0;6*8 b0L π 2

]a

Φ π2 : {xa ; ya } 7→ {−ya ; xa }.

¼* ;-|,- 861|* .3*L/86b08? b /7*L2J}*; b0L* x i a [a] = . ya j

]`

ðÍwÍñÍ rZ {)0/708? ;68302 70,*M,-:- -.*368-36 Φ b +6[0/* {i, j} 0 [6.0/68? /--8b*8/8b2J}2J O-3;272 ]Ú= ÏztxzZ ! ! ! Φ π2 :

xa ya

0 −1 1 0

7−→

π 2

xa ya

Z± TWµxXxXWx u³xTÍÎWW ³utuTuzÍ txyzuTÍ ³TW TxÿxXWW êÍwÍñZ ðÍwÍñÍ rZ 6, Lb* b*3I0, A(2; 1) 0 B(6; 3) 36b,-/8-3-,,*:- 83*2:-7?K

,016 ABC = 6M80 83*8?J *:- b*3I0,2\ */70 0[b*/8,-\ )8- -+N-L b*3I0, 83*2:-7?,016 b .-39L1* A → B → C → A 9b79*8/9 Lb0|*,0*; .- )6/-b-M /83*71* = 0/8*;6 1--3L0,68 .39;-2:-7?,69 = ¸¾

àxÿxXWxZ ;= 30/2,-1 Æ^= H-/1-7?12 ,6 B :*-;*830)*/1-M .7-/1-/80 .-7-|08*7?,; /)086*8/9 b36}*,0* .3-80b )6/-b-M /83*7K −→ → −→ A 10\ 8- −AC = Φ (AB) = ;**; AB = {4; 2} = 7*L-b68*7?,-\ −AC → = {(4 cos(− )−2 sin(− ) Å √ √ C 2 cos(− 3; 1 − 2 3} = ) + 4 sin(− )} = {2 + Æ^ 0/ H − → Ç = = -7?[29/? O-3;27-M r = r + AC \ ,6N-K L0; r = {4 + √3; 2 − 2√3} = ðÍwÍñÍ rZ -/86b08? 236b,*,09 836*18-300\ -.0/b6*;-M 8-)1-M M \ 7*|6}*M ,6 -132|,-/80 ω 36L02/6 R \1689}*M/9 +*[ /1-7?|*,09 .- L6,,-M .39;-M ` 017-0L6= àxÿxXWxZ ;= 30/2,-1 Æ= − π3

π 3

π 3

π 3

π 3

C

A

C

y

M

C t

O

x

A

Ç0/= Æ= {+*3*; -38-,-3;03-b6,,M 3*.*3 {O; i, j} ,6 .7-/1-/80 /7*L2J}0; -+36[-; [6 ,6)67- 3*.*36 b-[?;*; -L,- 0[ .-7-|*,0M 8-)10 M \ 1-:L6 -,6 -16[b6*8/9 ,6 .39;-M ` \ b 16)*/8b* b*18-36 i b-[?;*; *L0,0),M ,6.36b79J}0M b*18-3 .39;-M ` \ 0 +2L*; .3*L.-76:68?\ )8- b*18-3 j ,6K .36b7*, -8 .39;-M ` b 82 /8-3-,2\ :L* 36/.-7-|*,6 -132|,-/8?= H2/8? ) C · *,83 -132|,-/80 b O01/03-b6,,M ;-;*,8 b3*;*,0\ A · 8- 16 → −→ −−→ 321-b-LK 16/6,09 ω 0 ` \ 6 t = ∠ACM = ;**; r = −OA + AC + CM − → /8b2*;/9 .36b07-; [6;16,09 7-;6,-M®\ |OA| = | ^ AM | = Rt \ /7*K −→ H → −−→ L-b68*7?,-\ −OA = Rt i = 67**\ AC = R j \ CM = R e( − t) = -~8-;2 r = Rt i + R j + R e( − t) = {Rt − R sin t; R − R cos t} = ðÍwÍñÍ ±Z H- -132|,-/80 ω \[6L6,,-M 236b,*,0*; x +y = R \ 16808/9 +*[ /1-7?|*,09 .39;69 ` \ ,6)67?,-* .-7-|*,0* 1-8-3-M x = R = -/86b08? 236b,*,09 836*18-300\ -.0/b6*;-M 8-)1-M M \ 7*|6}*M ,6 ` \ .30,0;69 ¸Á M

3π 2

M

3π 2

2

2

2

[6 ,6)67?,-* ** .-7-|*,0* 8-)12 M (R; 0) ~b-7?b*,86 -132|,-/80= àxÿxXWxZ ;= 30/2,-1 Æ]= H2/8? A y · 8-)16−→16/6,09 ω 0 −`−\→6 t = ∠M OA = ;**; OA = R e(t) \ |AM | = | ^ AM | = −−→ Rt \ /7*L-b68*7?,-\ AM = Rt e(t − ) = → −−→ H-~8-;2 r = −OA + AM = R e(t) + 0

0

0 π 2

M

Rt e(t − Rt cos t}

π 2)

= {R cos t + Rt sin t; R sin t −

= ðÍwÍñW W S³TÍXxXW´á â]ä \ Ú_\ Úã\ Ú`\ ò\ ò]\ òÆ\ Æã]\ ÆãÚ\ Æa]\ Æa^\ ÆaÆ\ ÆaÚ Å âÚä \ 8*;6 `=

A O

M

t

M0

x

Ç0/= Æ]=

hohe mihpqeçekpe qefghihq kl mnhofhogp ' Ï³TxwxxXWxZ ¡¢ª a b ¦ ¡ ¢£ ªÀ« ¤ !

ha, bi = ([a], b).

{ /0/8*;* 1--3L0,68\ -.3*L*79*;-M .36b; -38-,-3;03-b6,,; 3*.*K 3-; {O; i, j} \ 2)08b69 `\ .-72)6*; x y a a ha, bi = −ya xb + xa yb = . xb yb

VxvTÍWñx¶yWx ¶tuÌ¶ztÍ yu¶uVu ³TuWêtxwxXW´Z

Æ^

1-/-/0;;*830),-/8?= +070,*M,-* -8-+36|*,0* = · 79 L-16[68*7?/8b6 ~80N /b-M/8b L-/868-),- b-/.-7?[-b68?/9 070 1--3K L0,68,-M O-3;27-M Æ^ 070 /b-M/8b6;0 /16793,-:- .3-0[b*L*,09 0 -.*K 3600 .-b-3-86 ,6 2:-7 = [ 1 \ b )6/8,-/80\ /7*L2*8\ )8- ha, ai = 0 = 1◦ ha, bi = −hb, ai 2◦ ε : E2 × E2 3 {a, b} 7→ ha, bi ∈ R π 2

◦

¸È

H-7?[29/? /b-M/8b6;0 1 0 2 \ 7*:1- .-72)08? O-3;272 L79 b)0/7*K ,09 1-/-:- .3-0[b*L*,09 b .3-0[b-7?,-; 6OO0,,-; 3*.*3* {O; e } ,6 E ha, bi = ha e , b e i = a b he , e i = ε a b = ε a b + ε a b \ :L* ε = he , e i = 610; -+36[-;\ ◦

◦

2

i

i

i

i

j

j

i j

i

j

ij

i j

12

1 2

21

2 1

ij

j

a1 a2 ha, bi = ε12 1 2 . b b

xuµxzTWñx¶yWx ¶tuÌ¶ztÍ yu¶uVu ³TuWêtxwxXW´Z

79 1-/-:- .3-0[b*L*,09 0;**8/9 /7*L2J}69 O-3;276\ 6,67-:0),69 O-3;27*\ .- 1-8-3-M -.3*L*79*8/9 /16793,-* .3-0[b*L*,0* b*18-3-b Æ ha, bi = |a||b| sin ϕ, :L* ϕ = ϕ − ϕ · 2:-7 a ª b = *M/8b08*7?,-\ b

a

ha, bi = ([a], b) = |a||b| cos(ϕa +

π − ϕb ) = |a||b| sin(ϕb − ϕa ). 2

[ O-3;27 Æ b8*16J8 /7*L2J}0* /b-M/8b6 1-/-:- .3-0[b*L*,09 |ha, bi| · .7-}6L? .63677*7-:36;;6\ .-/83-*,,-:- ,6 b*18-36N a 0 b = ] ha, bi > 0 ⇐⇒ {a, b} · .36bM +6[0/= ha, bi < 0 ⇐⇒ {a, b} · 7*bM +6[0/ = bi = 0 ⇐⇒ a||b = ;**8 Æ ha, ;*/8- /7*L2J}69 O-3;276 L79 .7-}6L0 83*2:-7?,016 ABC ,6 .7-/1-/80 S4ABC

x −x y −y 1 −→ −→ 1 B A B A = |h AB, AC i| = mod xC − xA yC − yA 2 2

.

→ −→ Ú 79 83*2:-7?,016 ABC ,6 .7-/1-/80 h −AB, AC i > 0 \*/70 -+N-L b*3K I0, A 7→ B 7→ C 7→ A .3*L/86b79*8 /-+-M Lb0|*,0* b-132: 83*2:-7?,016 → −→ b .-7-|08*7?,-; ,6.36b7*,00 .3-80b )6/-b-M /83*710\0 h −AB, AC i < 0 \ */70 -+N-L b*3I0, A 7→ B 7→ C 7→ A 9b79*8/9 Lb0|*,0*; .- )6/-b-M /83*71* =

¸Ô

ðÍµxñÍXW´Z =6 -30*,803-b6,,-M *b170L-b-M .7-/1-/80 E .63677*7-:36;; ABCD = −→ −−→ P (AB, AD) " [6L6,,; ,6.36b7*,0*; -+N-L6 b*3I0, A 7→ B 7→ C 7→ D 7→ A ,6[b6*8/9 .63677*7-:36;;-; = >0/7−→ −−→ Æ] Se = h AB, AD i .30 ~8-; ,6[b6J8 ¦ 070 ¦ .7-}6L?J -30K *,803-b6,,-:- .63677*7-:36;;6 ABCD = ]= 5,67-:0),-\ L79 © 83*2:-7?,016 ABC \ 8- */8? 83*K 2:-7?,016 / [6L6,,; ,6.36b7*,0*; -+N-L6 b*3I0,\ ,6 -30*,803-b6,,-M .7-/1-/80 E -.3*L*7*,- )0/71 −→ −→ ÆÆ Se = h AB, AC i, 2 ,6[b6;-* ¦ 070 ¦ .7-}6L?J ~8-:- -30*,K 803-b6,,-:- 83*2:-7?,016= ðÍwÍñÍ ±rZ H3-b*308?\ )8- [,6)*,09 -30*,803-b6,,N .7-}6L*M Æ] 0 ÆÆ ,* ;*,9J8/9 .30 0170)*/1-M .*3*/86,-b1* b*3I0, A 7→ B 7→ C 7→ D 7→ A 0 A 7→ B 7→ C 7→ A /--8b*8/8b*,,- = ðÍwÍñÍ ±±Z 6M80 b/-82 h 83*2:-7?,016 ABC / b*3I0,6;0 A(1; −2) \ B(4; 2) \ C(2; 4) = àxÿxXWxZ h = 2S /|−AB| → 3 4 = /5 = = 1 6 ðÍwÍñW W S³TÍXxXW´á â]ä \ `a\ ``\ ^\ ^]\ ^ÆÅ âÚä \ 8*;6 ^= # $il npkp kl lpkkh% mnhofhogp ' Zr àÍêWñXUx tWwU STÍtXxXWÌ ³T´µuÌ XÍ Í

WXXuÌ ³u¶yu¶zWZ H39;69 ` ,6 6OO0,,-M .7-/1-/80 A -L,-[,6),- -.3*L*79*8/9 .3-0[b-7?K ,-M /b-*M 8-)1-M M ∈ ` 0 -L,-;*3,; ,6.36b79J}0; .-L.3-/836,/8b-; { V (`) \ /-/8-9}0; 0[ b*18-3-b .63677*7?,N ` = /910M ,*,27*b-M b*18-3 H a ∈ V (`) ,6[b6*8/9 ¡£À« .39;-M ` = 30 ~8-; .636 {M ; a} 9b79*8/9 6OO0,,; 3*.*3-; ,6 .39;-M ` \ 36//;6830b6*;-M 161 -L,-;*3,-* 6OO0,,-* .3-/836,/8b- = ¸Ù 2

ABCD

2

4ABC

c

c

14 5

4ABC

2

0

1

1

0

H2/8? r · 36L02/Kb*18-3 8-)10 M \ 6 r · 36L02/Kb*18-3 .3-0[b-7?,-M 8-)10 M ∈ A .- -8,-I*,0J 1 3*.*32 {O; e , e } = {O; e } \ i = 1, 2 = 0

0

2

1

M0

;**; M

i

M

a

O

2

Ç0/= ÆÆ=

L79 ,*1-8-3-:= 8/JL6 .-72)6*; 236b,*,09 .39;-M\ ,6[b6*;*

−−−→ ∈ ` ⇐⇒ M0 M ∈ V1 (`) ⇐⇒ t ∈ R ⇐⇒ r − r0 = ta

¡¤

−−−→ M0 M = ta

ÆÚ )8- b 236b,*,09N ÆÚ .636;*83 t .3*L/86b79*8 /-+-M 1--3L0,682 8;*80;\ 8-)10 M b 3*.*3* {M ; a} ,6 .39;-M ` = /17J)69 .636;*83 t 0[ 236b,*,0M ÆÚ\ .-72)6*; 861 ,6[b6*;-* º ¤ 236b,*,0* .39;-M r = r0 + ta ⇐⇒ xi = xi0 + tai , i = 1, 2. 0

x1 − x10 x2 − x20 = . a1 a2

(36b,*,0* .39;-M ` ;-|,- .-72)08? 861|*\ [6.0/b69 2/7-b09 70,*MK ,-M [6b0/0;-/80 b*18-3-b r − r 0 ta x −x x −x Æò = 0. 0

1

2

1 0

a1

2 0

a2

-.3*L*708*7? b 236b,*,00 Æò\ .-72)6*; a (x − x ) − a (x − Ç6/13b69 ) x ) = 0 \ 8- .-[b-79*8 [6L68? .39;2J ` 236b,*,0*; .*3b-M /8*.*,0 Æ_ A x + A x + A = 0, :L* A = a \ A = −a = (36b,*,0* Æ_ ,6[b6J8 ¬« ¡£º ¦ ` = TxwuxXWxZ ¡¦ ¡ (36) (A + A 6= 0) ¢ª ¡£À A ¡£À« a = {−A ; A } = ¾Ý 2 0

1

1

2

2

1

2

2

2

1

3

1

2 1

2

1 0

2

2 2

1

1

2

éuyÍêÍzxë¶ztuZ H2/8? (x ; x ) · .3-0[b-7?,-* 3*I*,0* 236b,*,09 Æ_\ 8- */8?\ A x + A x + A = 0 = {)0869 ~8- /--8,-I*,0* 0[ Æ_\ .-72)0; ) A (x − x ) + A (x − x ) = 0 \ 8- 9b79*8/9 236b,*,0*; .39;-M ` \ 0;*JK }*M ,6.36b79J}0M b*18-3 a = {−A ; A } 0 .3-N-L9}*M )*3*[ 8-)12 M / 1--3L0,686;0 (x ; x ) = sxw¶ztW´Z è a ¡ ¡£¦ ` ¥ ¢ª¦ (36) ¥ ©ª ©ª¥ ©ª 1 0

1

1

1 1 0 1 0

2 2 0 2 2

2 0

3 2 0

2

1 0

1

0

2 0

A1 a1 + A2 a2 = 0.

££ ` ¥ ¢ª£ (36) ¥ ¡ ª¦ ª ª ª Ox © © ¥ © A = 0 § ££ ` ¥ ¢ª£ (36) ¥ ¡ ª¦ ª ª ª Ox © © ¥ © A = 0 = H2/8? Lb* .39;* ` 0 ` [6L6,\ /--8b*8/8b*,,-\ 236b,*,09;0 A x + A x + A = 0 0 A x + A x + A = 0 = -:L6 H39;* ` 0 ` .63677*7?, b /83-:-; /;/7* /7-b6 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 1

1

2

2

0

2

2

0 1 1

3

1

0 2 2

1

0 3

0

A01 A02 A03 = 6= . A1 A2 A3

H39;* ` 0 ` /-b.6L6J8 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 0

A01 A02 A03 = = . A1 A2 A3

H39;* ` 0 ` .*3*/*16J8/9 b -L,-M 8-)1* 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 0

A01 A02 6= . A1 A2

óTÍtXxXW´ ³T´µuÌ & ³Tuõuw´'xÌ ñxTxê wtx wÍXXUx zuñyWZ 79 /-/86b7*K ,09 £ ¡£¦ ` ¥ ¡Ðª£«¦ ¤¢ ª ¤ A(x ; x ) 0 B(x ; x ) \ → [6 ,6.36b79J}0M b*18-3 L-/868-),- b[98? 8-)12 A [6 M \ 6 b*18-3 −AB a= { 3*[27?868* .-72)0; /7*L2J}0* 236b,*,09 1 A

2 A

1 B

0

r = rA + t(rB − rA )

⇐⇒

x2 − x2A x1 − x1A = 2 x1B − x1A xB − x2A

xi = xiA + t(xiB − xiA ), i = 1, 2, ⇐⇒ x1 − x1 x2 − x2 A A ⇐⇒ 1 = 0. 1 2 xB − xA xB − x2A

¾Ü

2 B

óTÍtXxXWx ³T´µuÌ t uzTxêyÍõZ ¼/70 0[b*/8, 8-)10 A(a; 0) 0 B(0; b) .*3*/*)*,09 .39;-M ` / -/9;0 1--3L0,68 Ox 0 Ox a 0 b · -83*[10\ -8/*16*;* .39;-M ` ,6 -/9N 1--3L0,68\ 8- 236b,*,0* .39;-M ` ;-|,[6.0/68? b b0L* x x Æã + = 1. a b (36b,*,0* Æã ,6[b6*8/9 ¡£¦ ¢Ð= Z± YêÍWµXux TÍ¶³uuxXWx wtSõ zuñxy uzXu¶WzxëXu ³T´µuÌ Z H2/8? [6L6,6 .39;69 ` / 236b,*,0*; A x + A x + A = 0 0 .636 8-)*1 { P (x ) 0 Q(x ) \ ,* 7*|6}0N ,6 .39;-M ` = b*L*; O2,10J α : R → R \ .-76:69 α(x , x ) = A x + A x + A = TxwuxXWxZ ¨¤ P (x ) Q(x ) » ¡ ª ¡£¦ ª ª ª ` © © ¥ © sgn α(x ) = sgn α(x ) = éuyÍêÍzxë¶ztuZ H3*L.-7-|0; /,6)676\ )8- P 0 Q 7*|68 .- 36[,* /8-3-, -8 .39;-M ` = H2/8? K · 8-)16 .*3*/*)*,09 .39;N ` 0 P Q = -:L6 = 61 161 K ∈ ` \ 8λ = (P QK) > 0 0 x = 1

2

1

1

i P

2

1

2

2

3

2

i Q

1

2

1

1

2

2

3

i P

i Q

i P

i Q

xiP +λxiQ 1+λ

i K

x2P + λx2Q x1P + λx1Q + A2 + A3 = 0, A1 1+λ 1+λ

)8- ~1b0b67*,8,- 8-;2\ )8- α(x ) + λα(x ) = 0 \ -812L6 − = λ > 0 \ /7*L-b68*7?,-\ sgn α(x ) = −sgn α(x ) = H2/8? 8*.*3? P 0 Q 7*|68 .- -L,2 /8-3-,2 -8 ` \ 0 S · ,*1-8-369 8-)K 16\ 7*|6}69 .- L32:2J /8-3-,2 -8 ` = -:L6 sgn α(x ) = −sgn α(x ) = sgn α(x ) = ðÍwÍñÍ ±ÊZ -/86b08? 236b,*,0* .39;-M\.3-N-L9}*M )*3*[ 8-)12 M (x ; x ) .63677*7?,- L6,,-M .39;-M ` / 236b,*,0*; A x + A x + A = 0 = àxÿxXWxZ )*b0L,-\ )8- 0/1-;-* 236b,*,0* 0;**8 b0L i P

i P

α(xiP ) α(xiQ )

i Q

i Q

i P

i S

i Q

1 0

1

1

2

2

3

A1 (x1 − x10 ) + A2 (x2 − x20 ) = 0.

ðÍwÍñÍ ±Z .3*L*708? .-7-|*,0* .39;-M ` \ 0;*J}*M 236b,*,0* x − 7x +5 = 0 \-8,-/08*7?,- 83*2:-7?,016 ABC / b*3I0,6;0 A(3; 1) \ B(−2; 4) \ C(1; 0) = ¾ï 1

2

2 0

àxÿxXWxZ ;= 30/2,-1 ÆÚ = ;**; α(x ) = 1 > 0 \ α(x ) = −25 < 0 \ H α(x ) = 6 > 0 = 7*L-b68*7?,-\ ` .*3*/*16*8 /8-3-, AB 0 BC = 2/8? K · 8-)16 .*3*/*)*,09 ` / .39;-M AC = 61 161 .3-/8-* -8,-I*,0* i A

i B

i C

−−→ AK α(xiA ) 1 (ACK) = −−→ = − = − , i ) 6 α(x KC C

) −→ −−→ 8- |−AK| < |KC| \ .-~8-;2 ` .*3*/*16*8 .3-L-7|*,0* /8-3-, AC [6 8- K 12 A = B

K

`

A

C

Ç0/= ÆÚ = àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*100 ò\ _Å âä \ :76b6 @\ åòÅ âòä \ :76b6 ò\ å= ðÍwÍñW W S³TÍXxXW´á â]ä \ ãÆ\ ãÚ\ ãa\ a\ aã\ a`\ `\ `]\ `Æ\ `Ú\ `ò\ `_\ `ã\ `a\ ]^^\ ]^\ ]^]\ ]^_\ ]^ã\ ]^`\ ]Æ_\ ]Æã\ ]Æa\ ]Ú_\ ]ÚãÅ âÚä \ 8*;6 ]= ZÊ óTÍtXxXWx ³SñyÍ ³T´µUõ W xVu ³TWµxXxXWxZ H2/8? L6, Lb* .*3*/*16J}0*/9 36[70),* .39;* ` 0 ` \ [6L6,,*\ /--8b*8/8b*,,-\ 236b,*,09;0 A x + A x + A = 0 0 B x + B x + B = 0 = Ç*I*,0* x = x ; x = x /0/8*; 236b,*,0M 1

1

2

1 0

1

2

2

3

1

1 1

2

2

2

3

2 0

(

A1 x1 + A2 x2 + A3 = 0 B1 x1 + B2 x2 + B3 = 0,

1-8-3-* b L6,,-; /72)6* /2}*/8b2*8 0 *L0,/8b*,,-\-.3*L*79*8 8-)12 M (x ; x ) .*3*/*)*,09 ` 0 ` = H30 7J+N λ 0 µ \ ,* 36b,N ,27J -L,-b3*;*,,-\ 236bK ,*,0* Æa λ(A x + A x + A ) + µ(B x + B x + B ) = 0 ¾ 1 0

1

2

1

1

2

2

3

1

1

2

2

3

2 0

[6L6*8 ,*1-8-32J .39;2J ` \.3-N-L9}2J )*3*[ 8-)12 M (x ; x ) = H-/1-7?12 ,6.36b79J}0M b*18-3 .39;-M ` 9b79*8/9 70,*M,-M 1-;+0,60*M λa + µb ,6.36b79J}0N b*18-3-b a = {−A ; A } 0 b = {−B ; B } .39;N ` 0 ` \ -+36[2J}0N +6[0/ b*18-3,-:- .3-/836,/8b6 V \ 6//-003-b6,,-:- / A \ 236b,*,0* b/91-M .39;-M\ .3-N-L9}*M )*3*[ 8-)12 M (x ; x ) \ ;-|,- .3*LK /86b08? b b0L* Æa .30 ,*1-8-3N λ 0 µ = -b-12.,-/8? .39;N\ .3-N-L9}0N )*3*[ -L,2 8-)12 M (x ; x ) \ ,6[b6K *8/9 ¡¤ ¡£Ð= -)16 M (x ; x ) .30 ~8-; ,6[b6*8/9 Ñ ¡¤= )*b0L,-\ .2)-1 .39;N / *,83-; M (x ; x ) [6L6*8/9 236b,*,0*; 2 0

1 0

2

1

2

1

1

2

2

2

1 0

2 0

2 0

1 0

1 0

2 0

1 0

2 0

A1 (x1 − x10 ) + A2 (x2 − x20 ) = 0.

¼/70 ` 0 ` · Lb* 36[70),* .39;*\ .30,6L7*|6}0* ,*1-8-3-;2 .2)12\ 8- .3-0[b-7?,69 .39;69 .2)16 0;**8 236b,*,0* Æa .30 ,*1-8-3N λ 0 µ \ .-~8-;2 236b,*,0* Æa · ~8- 236b,*,0* .2)16 .39;N = H2/8? 830 .39;* ` \ ` 0 ` \ /3*L0 1-8-3N 0;*J8/9 Lb* ,*.6367K 7*7?,*\ [6L6,\ /--8b*8/8b*,,-\ 236b,*,09;0 A x + A x + A = 0 \ B x + B x + B = 0 0 C x + C x + C = 0 = 80 830 .39;* .30,6L7*K |68 ,*1-8-3-;2 .2)12\ 8- */8?\ .*3*/*16J8/9 b -L,-M 8-)1*\ 8-:L6 0 8-7?18-:L6\ 1-:L6 /-b;*/8,6 /0/8*;6 236b,*,0M 1

2

1

2

3

1

1

1

2

2

3

1

1

2

2

1

2

2

3

3

 1 2   A1 x + A2 x + A3 = 0 B1 x 1 + B2 x 2 + B3 = 0   C1 x1 + C2 x2 + C3 = 0 .

Æ`

(/7-b0*; /-b;*/8,-/80 /0/8*; Æ` 9b79*8/9 36b*,/8b- ,27J -.3*L*708*K 79 -/,-b,-M ;6830 ~8-M /0/8*; A1 A2 A3 B1 B2 B3 = 0. C1 C2 C3

¼/70 .39;* ` 0 ` / 236b,*,09;0 A x +A x +A = 0 0 B x +B x + B = 0 .63677*7?, ,- ,* /-b.6L6J8\ 8- b/9169 .39;69\ 0;*J}69 236bK ,*,0* Æa .30 ,*1-8-3N λ 0 µ \ .63677*7?,6 ` 0 ` = * 832L,- .-16[68?\ )8- b ~8-; /72)6* b/9169 .39;69\ .63677*7?,69 ` 0 ` \ ;-|*8 +8? [6L6,6 1

2

1

1

2

2

3

3

1

2

1

¾¸

2

1

1

2

2

236b,*,0*; Æa .30 ,*1-8-3N λ 0 µ = {/J /-b-12.,-/8? .39;N Æa .30 ~8-; 861|* ,6[b6J8 .2)1-; ,*/-+/8b*,,; .39;N = 7*L2J}0* .30;*3 .-16[b6J8\ 161 .2)10 ;-:28 ~OO*180b,- .30;*K ,98?/9 .30 3*I*,00 :*-;*830)*/10N [6L6)= ðÍwÍñÍ ±Z -/86b08? 236b,*,0* .39;-M\ .3-N-L9}*M )*3*[ 8-)12 .*3*K /*)*,09 .39;N x +2x +5 = 0 0 2x −3x −2 = 0 0 .63677*7?,-M .39;-M 8x + 2x + 7 = 0 = àxÿxXWxZ (36b,*,0* 0/1-;-M .39;-M ` ;-|,- [6.0/68? b b0L* 1

1

2

1

2

2

λ(x1 + 2x2 + 5) + µ(2x1 − 3x2 − 2) = 0 ⇐⇒ (λ + 2µ)x1 + (2λ − 3µ)x2 + (5λ − 2µ) = 0.

61 161 ` .63677*7?,6 .39;-M 8x + 2x )+ 7 = 0 \ 8- = \ -812L6 λ = 2 \ µ = 1 λ 0 µ /2}*/8b*,, / 8- ,-/8?J L- .3-.-30-,67?,-/80 λ : µ 0\ /7*L-b68*7?,-\ 0/1-;69 .39;69 0;**8 236b,*,0* 4x + x + 8 = 0 = ðÍwÍñÍ ± Z *-3*;6 >*b 6 /8-3-,6N AB \ BC 0 CA 83*2:-7?,016 ) ) ABC L6, 8- 10 C \ A 0 B 8610*\ 8- (BCA ) = λ \ (CAB ) = λ 0 ) (ABC ) = λ = -16[68?\ 8- .39;* AA \ BB 0 CC .30,6L7*|68 -L,-K ;2 .2)12 8- */8?\ .*3*/*16J8/9 b -L,-M 8-)1* 070 .63677*7?, 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 λ λ λ = 1 = 1

2λ−3µ 2

λ+2µ 8

2

1

0

0

0

0

3

0

0

0

2

1

0

2

0

1 2 3

B0 C0

C

C A

A0

A0 A

B

C0

B0

B

Ç0/= Æò= àxÿxXWxZ ;= 30/2,-1 Æò= Ç6//;-830; 6OO0,,M 3*.*3 {O; e } \ :L* −→ −→ { ~8-; 3*.*3* b*3I0, 83*2:-7?,016 0 O = A \ e = AB \ e = AC = 36//;6830b6*;* ,6 *:- /8-3-,6N 8-)10 0;*J8 /7*L2J}0* 1--3L0,68 ¾¾ i

1

2

H \ \ \ \ \ ; ; 0 = -7?[2K A(0; 0) B(1; 0) C(0; 1) A B 0; C 9/? 236b,*,0*; .39;-M b -83*[16N\ .-72)6*;\ /--8b*8/8b*,,-\ /7*L2J}0* 236b,*,09 .39;N AA \ BB 0 CC λ1 1 1+λ1 1+λ1

0

0

0

λ2 1 x1 − x2 = 0, 1 + λ2 1 + λ2

0

1 1+λ2

0

λ3 1+λ3

0

1 + λ1 1 x + x2 = 1. λ1

x1 + (1 + λ3 )x2 = 1,

.30,6L7*|68 -L,-;2 .2)12 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 80 830 .39;* λ −1 0 2 1 + λ3 −1 = 0 1 1 + λ1 λ1 −λ1

⇐⇒

λ1 λ2 λ3 = 1.

àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*109 ]Ú Å â]ä \ :76b6 @@@=Å âòä \ :76b6 ò\ å= ðÍwÍñW W S³TÍXxXW´á â]ä \ ]ò\ ]ò]\ ]òÆ\ ]òÚ\ ]òò\ ]òãÅ âÚä \ 8*;6 Æ= ( $il npkp kl eqfnpçhqh% mnhofhogp ' Zr ²uTµÍëXUÌ txyzuT ³T´µuÌ Z

6 *b170L-b-M .7-/1-/80) E -L,-;*3,-* ,6.36b79J}** .-L.3-/836,/8bV (`) .39;-M ` -L,-[,6 ,- -.3*L*79*8/9 -L,-;*3,; .-L.3-/836,/8b-; { V (`) ⊂ E \ /-/8-9}0; 0[ b*18-3-b\ -38-:-,67?,N .39;-M ` = /910M ,*,27*b-M b*18-3 0[ .-L.3-/836,/8b6 V (`) ,6[b6*8/9 º .39;-M ` = óTÍtXxXWx ³T´µuÌ & ³Tuõuw´'xÌ ñxTxê zuñyS M W Wµx'xÌ XuTµÍë) XUÌ txyzuT N Z H2/8? r · 36L02/Kb*18-3 8-)10 M \ 6 r · 36L02/Kb*18-3 .3-0[b-7?,-M 8-)10 M ∈ E .- -8,-I*,0J 1 6OO0,,-;2 3*.*32 {O; e } \ −−−→ −−−→ i = 1, 2 = -:L6 M ∈ ` ⇐⇒ M M ⊥ N ⇐⇒ (M M , N) = 0 ⇐⇒ Ú^ (r − r , N) = 0. 2

1 ⊥ 1

2

⊥ 1

0

0

0

2

i

0

0

0

V1⊥ (`) N V1 (`)

a M0

M

Ç0/= Æ_=

¾Á

{ 1--3L0,686N 236b,*,0* Ú^ .30,0;6*8 b0L

Ú

gij (xi − xi0 )N j = 0.

¼/70 {O; e = i, e = j} · -38-,-3;03-b6,,M 3*.*3\ 8- g = (e , e ) = δ /0;b-7 P3-,*1*36\ 0 b 1--3L0,686N N{A; B} \ M (x; y) \ M (x ; y ) 236b,*K ,0* Ú .30,0;6*8 b0L Ú] A(x − x ) + B(y − y ) = 0. Ç6/13b69 /1-+10 b Ú]\ .30N-L0; 1 -+}*;2 236b,*,0J ÚÆ Ax + By + C = 0 .39;-M ` b .39;-2:-7?,-M /0/8*;* 1--3L0,68= P61 /7*L/8b0*\ .-72)6*; /7*L2J}** TxwuxXWxZ ö£ ¡£¦ ` ¥ À«¦ (43) º ¡¥ N{A; B} £££ ¥ º a{−B; A} ¡£À« = ðÍµxñÍXWxZ H636 )0/*7 {A ; A } 0;**8 :*-;*830)*/10M /;/7 0 L79 .39K ;-M b 6OO0,,-; .3-/836,/8b*\ 6 0;*,,-\ -,6 9b79*8/9 .63-M 1--3L0,68 70,*M,-M O2,100 ξ : V → R ,6 b*18-3,-; .3-/836,/8b* V = { /72)6* *b170L-b-M .7-/1-/80 ~86 O2,109 ξ 0;**8 b0L ξ(v) = (N, v) = H-L3-+,** ~8-8 b-.3-/ +2L*8 -+/2|L68?/9 .30 36//;-83*,00 .39;N 0 .7-/1-/8*M b 83*N;*3,-; .3-/836,/8b* = Z± àÍ¶¶zu´XWx uz zuñyW wu ³T´µuÌ Z { [6b0/0;-/80 -8 8-:-\ 161 [6L6,6 .39;69 ` ,6 *b170L-b-M .7-/1-/80 E \ 36//8-9,0* dist (M , `) -8 8-)10 M ∈ E L- ` ;-|*8 +8? ,6ML*,- /7*L2JK }0;0 /.-/-+6;0 M H3*L.-7-|0;\ )8- .39;69 ` [6L6,6 1. 8-)1-M M ∈ ` 0 ,6.36b79J}0; b*18-3-; a{x ; y } = -:L6 36//8-9,0* dist (M , `) ,6N-K L08/9 161 b/-86 .63677*7-:36;;6\ .-/83-K M a ` −−−→ /; 30/ Æã *,,-:,6 b*18-36N 0 Æã = = a MM 0/ Ç = = ¾È 1

2

ij

0

0

1

i

0

j

ij

0

0

2

2

2

2

1

1

2

•

0

a

1

a

•

0

0

1

x −x y −y 1 0 1 0 mod −−−→ xa ya |h M0 M1 , a i| p = dist (M1 , `) = |a| x2a + ya2

ÚÚ ) H ) 2. 3*L.-7-|0;\ 8- .39;69 ` [6L6,6 8- 1-M M ∈ ` 0 ,-3;67?,; b*1K 8-3-; N / 1--3L0,686;0 {A; B} = -:L6 36//8-9,0* dist (M , `) ,6N-L08/9 161 6+/-7J8,69 b*70)0,6 .3-*100 b*18-36 −M−−M→ ,6 -/? / ,6.36b79J}0; b*18-3-; N /;= 30/= Æa .

0

1

0

1

−−−→ dist (M1 , `) = |prN (M0 M1 )| =

−−−→ |(M0 M1 , N)| |A(x1 − x0 ) + B(y1 − y0 )| √ . = = |N| A2 + B 2

¼/70 .39;69 [6L6,6 236b,*,0*; ÚÆ\8- O-3K ;272 Úò ;-|,- .*3*.0/68?\36/13b69 /1-+K 10\ b b0L* dist (M1 , `) =

Úò

N •

|Ax1 + By1 + C| √ . A2 + B 2

M

Æa= ` 0/ Ç = { 6OO0,,-M /0/8*;* 1--3L0,68\ -.3*L*K 79*;-M 6OO0,,; 3*.*3-; ,6 E \ O-3;27 ÚÚ 0 Úò .30,0;6J8\ /--8K b*8/8b*,,-\ b0L •

M0

2

0

x1 − x1 x2 − x2 |ε12 |mod 1 1 0 1 2 0 a a |εij (xi1 − xi0 )aj | p p = dist (M1 , `) = gij ai aj gij ai aj |gij (xi1 − xi0 )N j | dist (M1 , `) = p . gkm N k N m

ZÊ óVu µxwS wtSµ´ ³T´µUµW WXW´µWZ

Ú_ Úã

(:7-; ;*|L2 -30*,803-b6,,;0 .39;;0 ` ↑ 0 ` ↑ ,6[b6*8/9 2:-7 ϕ ∈ [0, π] ;*|L2 0N ,6.36b79J}0;0 b*18-36;0 a 0 a = 8-8 2:-7 ,6N-L08/9 ¾Ô 1

2

1

2

.- O-3;27* cos ϕ =

(a1 , a2 ) . |a1 ||a2 |

H2/8? 8*.*3? ` 0 ` · ,*-30*,803-b6,,* .39;* / ,6.36b79J}0;0 b*18-36;0 a 0 a /--8b*8/8b*,,- = H2/8? ϕ · 2:-7 ;*|L2 a 0 a \ 6 ϕ · 2:-7 ;*|L2 −a 0 a = L0, 0[ ~80N 2:7-b .30,6L7*|08 0,8*3b672 [0, ] \ -, 0 9b79*8/9 .- -.3*L*7*,0J 2:7-; ;*|L2 .39;;0 ` 0 ` = (:-7 ϕ ;*|L2 .39;;0 ` 0 ` ,6N-L08/9 .- O-3;27* 1

1

2

2

1

1

1

2

π 2

2

1

1

2

2

2

cos ϕ =

|(a1 , a2 )| . |a1 ||a2 |

¼/70 0[b*/8, ,-3;67?,* b*18-3 N 0 N ~80N .39;N\8- 2:-7 ϕ ;-|,,6N-L08? 861|* .- O-3;27* 1

cos ϕ =

2

|(N1 , N2 )| . |N1 ||N2 |

{ /72)6*\*/70 ` 0 ` [6L6,\/--8b*8/8b*,,-\236b,*,09;0 A x+B y+C 0 0 A x + B y + C = 0 \ 81

2

2

2

1

1

1

=

2

|A1 A2 + B1 B2 | p . cos ϕ = p 2 A1 + B12 A22 + B22

Z TWµxTUZ ðÍwÍñÍ ±Z -/86b08? 236b,*,0* .*3.*,L0127936 1 .39;-M 5x − 2y + 3 = 0 \

.3-N-L9}*:- )*3*[ 8-)12 M (2; 3) = àxÿxXWxZ 6.36b79J}0; b*18-3-; 0/1-;-:- .*3.*,L0127936 9b79*8/9 b*18-3 / 1--3L0,686;0 {5; −2} = H-~8-;2 .*3.*,L012793 0;**8 /7*L2J}0* .636;*830)*/10* 236b,*,09 x = 2 + 5t Å y = 3 − 2t = 32:-M b6306,8 36//2|L*,0M ,-3;67?,; b*18-3-; 0/1-;-:- .*3.*,K L0127936 9b79*8/9 b*18-3 / 1--3L0,686;0 {2; 5} \ .-~8-;2 .*3.*,L012793 0;**8 236b,*,0* 2(x − 2) + 5(y − 3) = 0 = ðÍwÍñÍ ±Z -/86b08? 236b,*,0* +0//*1830/ 2:76 ∠A 83*2:-7?,016 ABC / b*3I0,6;0 A(3; 1) \ B(−2; 4) 0 C(1; 0) = àxÿxXWxZ [ 0[b*/8,-:- /b-M/8b6 L06:-,67*M 3-;+6 /7*L2*8\ )8- L79 8-:-\ )8-+ .-72)08? b*18-3\ L*79}0M .-.-76; 2:-7 ;*|L2 b*18-36;0 a ¾Ù

0 b \ L-/868-),- b[98? /2;;2 b*18-3-b\ 1-770,*63,N b*18-36; a 0 b 0 0;*J}0N -L0,61-b2J L70,2\ ,6.30;*3\ + 070 |b|a + |a|b = 610; -+36[-;\ +0//*1830/6 2:76 ∠A 0;**8 /7*L2J}** 236b,*,0* a |a|

b |b|

−→ −→ −→ −→ r = rA + (|AC|AB + |AB|AC) t .

ðÍwÍñÍ ±Z 6M80 36//8-9,0* -8 8-)10 M (2; 1) L- .39;-M ` \ [6L6,,-M 236b,*,0*; 2x − 3x − 5 = 0 b ,*1-8-3-M 6OO0,,-M /0/8*;* 1--3L0,68\ */70 0[b*/8, g = 4 \ g = 8 \ g = 25 = àxÿxXWxZ H2/8? ω · 2:-7 ;*|L2 e 0 e = -:L6 1

2

11

12

22

1

cos ω =

2

g12 (e1 , e2 ) 4 =√ √ = |e1 ||e2 | g11 g22 5

=⇒

sin ω =

3 . 5

.-72)6*; |ε | = |he , e i| = |e ||e | sin ω = 6 = 8/JL6 67**\ M (1; −1) ∈ ` \ b*18-3 −M−−M → 0;**8 1--3L0,68 {1; 2} \ ,6.36b79K J}0; b*18-3-; .39;-M ` 9b79*8/9 b*18-3 a{3; 2} \ -812L6 .- O-3;27* Ú_ ,6N-L0; 12

1

2

1

0

2

0

dist (M, `) = √

1 2 6 · mod 3 2

12 =√ . 4 · 9 + 2 · 8 · 6 + 25 · 4 58

>8-+ b-/.-7?[-b68?/9 O-3;27-M Úã\ /,6)676 ,2|,- ,6M80 ,-3;67?K ,M b*18-3 N{N ; N } .39;-M ` 0[ 236b,*,09 g a N = 0 ⇐⇒ 4 · 3 · N + 8 · 3 · N + 8 · 2 · N + 25 · 2 · N = 0 = ðÍwÍñÍ ÊZ -16[68?\ )8- b/-8 83*2:-7?,016 .*3*/*16J8/9 b -L,-M 8-)1* .30,6L7*|68 -L,-;2 .2)12 .39;N= àxÿxXWxZ {/-8 AA \ BB 0 CC 83*2:-7?,016 ABC 0;*J8\ /--8b*8K /8b*,,-\ 236b,*,09 −−→ −→ −→ Úa (r − r , BC) = 0, (r − r , CA) = 0 0 (r − r , AB) = 0. 0/8*;6 83*N 236b,*,0M (48) /-b;*/8,6\ .-/1-7?12\ 161 7*:1- .3-b*308?\ 0N /2;;6 */8? 8-|L*/8b- 0 = 0 = àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*109 òÅ âä \ :76b6 @\ åòÅ âòä \ :76b6 ò\ åå\ ]= ðÍwÍñW W S³TÍXxXW´á â]ä \ ]^\ ]\ ]]\ ]Æ\ ]Ú\ ]ã\ ]]Ú\ ]òa\ ]_Æ\ ]_Ú\ ]_ò\ ]__\ ]_ã\ ]_`\ ]ãò\ ]aÆ\ ]`\ ]`Æ\ Æ^^\ Æ^Å âÚä \ 8*;6 Ú = ÁÝ 1

1

2

2

ij

1

0

A

i

j

2

0

0

B

C

hkp+eofpe oe+ekp ' Ç6//;-830;Hb E Lb* .39;* ` 0 ` \ .*3*/*16J}0*/9 b 8-)1* S .-L 2:7-; α ∈ (0, ) = 39;2J ` [6O01/032*;\ 6 .39;2J ` +2L*; b36}68? b-132: H e b E \ ,6[b6*;2J = ` = 30 ~8-; .39;69 ` -.0I*8 .-b*3N,-/8? Φ H39;69 ` 9b79*8/9 -/?J /0;;*8300 1-,2/6 Φe 0 ,6[b6*8/9 À ~8-:- 1-K ,2/6= -)16 S ,6[b6*8/9 b*3I0,-M 1-,2/6= P-,2/ Φe .3*L/86b79*8 /-+-M -+*L0,*,0* .39;N\ .3-N-L9}0N )*3*[ b*3I0,2 S \ .-72)6J}0N/9 .-b-K 3-8-; ,6)67?,-M .39;-M ` = P6|L69 .39;69 ~8-:- /*;*M/8b6 ,6[b6*8/9 .39;-70,*M,-M ¬¢À«¦ 1-,2/6 Φe = °,-|*/8b- Φ =) Φe ∩ π \ .- 1-8-3-;2 1-,2/ Φe .*3*/*16*8/9 .7-/1-/8?J π \ ,* .3-N-L9}*M *3*[ *:- b*3I0,2 S \ ,6[b6*8/9 ¤ ¤= P-,0)*/1-* /*)*,0* Φ 9b79*8/9 .-L;,-|*/8b-; .7-/1-/80 π = {9/,0;\ )8.3*L/86b79*8 /-+-M ~8- .-L;,-|*/8b- Φ ⊂ π = +-[,6)0; )*3*[ β ∈ [0, ] 2:-7 ;*|L2 .7-/1-/8?J π 0)-/?J 1-,2/6= Ç6//;-830;Æ .- -8L*7?,-/80 /7*L2J}0* 830 b-[;-|,N /72 69 β > α \ ] β = α \ β < α = sSñÍÌ rø β > α Z H30 β = 1-,0)*/1-* /*)*,0* Φ = Φe ∩ π .3*L/86b79*8 /-+-M -132|K ,-/8?= 67** +2L*; .3*L.-76:68?\ )8- β < = /2}*/8b0; /7*L2J}0* .-K /83-*,09 /;= 30/= Æ`= {.0I*; b 1-,2/ Φe Lb* /O*3 Σ 0 Σ \ 16/6J}0*K /9 .7-/1-/80 π = -)10 16/6,09 /O*3 Σ 0 Σ / .7-/1-/8?J π -+-[,6)0;\ /--8b*8/8b*,,-\ )*3*[ F 0 F = P6|L69 0[ /O*3 Σ 0 Σ 16/6*8/9 1-,2K /6 Φe .- -132|,-/80\ 7*|6}*M b .7-/1-/80\ .*3.*,L012793,-M -/0 1-,2/6= )*3*[ ω 0 ω \ 6 .7-/1-/80 )*3*[ ρ 0 +-[,6)0; 216[6,,* -132|,-/80 ) ) ρ /--8b*8/8b*,,- = +-[,6 0; 861|* *3*[ D 0 D .39;*\ .- 1-8-3; .7-/1-/8? π .*3*/*16*8\ /--8b*8/8b*,,-\ .7-/1-/80 ρ 0 ρ = H2/8? M · .3-0[b-7?,69 8-)16 1-,0)*/1-:- /*)*,09 Φ = +-[,6)0; )*K 3*[ A 0 A 8-)10 .*3*/*)*,09 -+36[2J}*M SM / -132|,-/89;0 ω 0 ω /--8b*8/8b*,,- = +-[,6)0; [68*; )*3*[ C 0 C -/,-b6,09 .*3.*,L01279K 3-b\ -.2}*,,N 0[ 8-)10 M ,6 .39;* D 0 D \ 6 )*3*[ P 0 P -/,-b6,09 .*3.*,L012793-b\ -.2}*,,N 0[ 8-)10 M ,6 .7-/1-/80 ρ 0 ρ = *

3

π 2

1

2

1

1

2

2

3

1

2

π 2

π 2

π 2

1

1

1

2

2

1

1

2

2

2

1

1

2

1

1

2

2

2

1

1

1

ÁÜ

2

2

1

2

1

2

2

Ç0/= Æ`= H-/1-7?12 b/910* Lb* 16/68*7?,* 1 /O*3*\ .3-b*L*,,* 0[ -L,-M 0 8-M |* 8-)10\ 36b,\ 8- |−M−F→| = |−M−−A→| \ 6 |−M−F→| = |−M−−A→| = 8/JL6 /7*L2*8\ )81

1

2

2

−−→ −−→ −−−→ −−−→ −−−→ |M F1 | + |M F2 | = |M A1 | + |M A2 | = |A1 A2 |. −−−→ |A1 A2 | ω1 ω2 M ∈Φ M

\ .3*L/86b79J}69 /-+-M L70,2 -83*[16 -+36[2J}*M - b*70)0,6 0 \ ,* [6b0/08 -8 bK SM 1-,2/6\ 7*|6}*:- ;*|L2 -132|,-/89;0 +-36 8-)10 = 610;)-+36[-;\ 16|L69 8-)16 1-,0)*/1-:- /*)*,09 Φ b 36//;6830b6*;-; /72 6* 2L-b7*8b-39*8 /7*L2J}*;2 2/7-b0J −−→ −−→ Ú` |M F | + |M F | = const. Áï 1

2

.7-/1-/80\ 2L-b7*8b-39J}0N 2/7-b0J Ú`\ °,-|*/8b- 8-)*1 *b170L-b-M ,6[b6*8/9 Ã¡= -),**\ /O-3;27032*; /7*L2J}** -.3*L*7*,0* = Ï³TxwxxXWxZ ¡ E ª ª ¤ F F ¥ £º »ª 2c § ß¡ É F F ¢£ » Φ Ð ¤ M ¡¥ £¦ Ð ª F F ¡£ ¤ 2a > 2c ¥ ¥ −−→ −−→ ò^ Φ = {M ∈ E : |M F | + |M F | = 2a}. 8-),-/8?J L- .-7-|*,09 ,6 .7-/1-/80 ~770./ -.3*L*79*8/9 Lb2;9 .6K 36;*836;0 2a 0 2c = 8,-I*,0* ~80N .636;*83-b 2

1

1

1

2

2

2

2

1

e=

2

c <1 a

,6[b6*8/9 ÃÑ ~770./6= S 79 ~770./6\ 9b79J}*:-/9 1-,0)*K /10; /*)*,0*;\.636;*83 2a 36b*, L70K α ,* -83*[16 A A b,*I,*M -+}*M 16/6K 8*7?,-M .39;-M 1 /O*36; Σ 0 Σ \ 6 A O .636;*83 2c 36b*, L70,* -83*[16 F F b,283*,,*M -+}*M 16/68*7?,-M .39;-M F 1 ~80; /O*36;= H-/1-7?12 |−A−−A→| = −−−→ −−→ −−−→ β |O O | cos α \ 6 |F F | = |O O | cos β \ 8^ /;= 30/= Ú F cos β ò . e= cos α { /72)6*\1-:L6 O-12/ ~770./6 ò^ O /-b.6L6J8 F = F = O \ ~770./ .3*LK /86b79*8 /-+-M ;,-|*/8b- b/*N 8-)*1\ ,6N-L9}0N/9 ,6 36//8-9,00 a -8 8-)K 10 O \ 8- */8? » 36L02/6 a ^= 0/ Ú Ç = ) / *,83-; b 8- 1* O = 1/*,830/08*8 -132|,-/80 36b*, ,27J= H2/8? ~770./ Φ ,* 9b79*8/9 -132|,-/8?J 0 M ∈ Φ = >0/76 r 0 r = |−M−F→| ,6[b6J8/9 É ª 8-)10 M = Á 1

2

1

2

1

1

1 2

1

1

1

2

1 2

1

2

2

2

A2

2

1

2

2

2

1

−−→ = |M F1 |

[ 36//;-83*,09 .39;-2:-7?,N 83*2:-7?,01-b M P A 0 M P C L79 ~770./6 Φ \ 9b79J}*:-/9 1-,0)*/10; /*)*,0*; 30/= Æ`\ .-72)6*;\ )8) −−→ −−−→ −−→ −−−→ |M P | = |M C | cos β \ |M P | = |M A | cos α = 8/JL6 /7*L2*8\ 8−−−→ −−→ |M A | cos β |M F | ò] −−−→ = −−−→ = cos α = e. |M C | |M C | .39;-2:-7?,* 83*2:-7?,010 M P A 0 M P C \ .30L*; 1 Ç6//;6830b69 6,67-:0),-;2 /--8,-I*,0J −−−→ −−→ |M A | |M F | òÆ −−−→ = −−−→ = e. | |M C | H39;* D 0 D ,6[b6J8/9|MªC ~770./6 Φ = Ç6//8-9,09 -8 8-)K 10 M ~770./6 L- *:- L03*1830/ -+-[,6)6J8 /7*L2J}0; -+36[-; d = −−−→ −−−→ |M C | \ d = |M C | = [ .-72)*,,N /--8,-I*,0M ò] 0 òÆ b8*16*8 /7*L2J}** /b-M/8b~770./6= É© ª ¤ M Ã¡ £À ¿Ä Ã¦ ¤ ª À«¦ ª ¢ ¬ ¤º M ÃÑ Ã¡ r r òÚ = = e. d d ðÍµxñÍXWxZ {I*.30b*L*,,* 36//2|L*,09 -8,-/98/9 1 ~770./2\ 9b79K J}*;2/9 1-,0)*/10; /*)*,0*;= 0|* +2L*8 216[6,-\ 161 ,6N-L98/9 L03*1K 830/ .3-0[b-7?,-:- ~770./6\ -.3*L*7*,,-:- 2/7-b0*; ò^= ,ÍXuXWñx¶yux STÍtXxXWx -W³¶ÍZ Ç6//;-830; ~770./ Φ \)-.3*L*7*,,M 161 ;,-|*/8b- 8-)*1 ò^= [ ~8-K :- -.3*L*7*,09 /7*L2*8\ 8- .39;69 F F 0 .*3.*,L012793 1 F F \ L*79K }0M -83*[-1 F F .-.-76;\ 9b79J8/9 -/9;0 /0;;*8300 ~770./6= {+*3*; .39;-2:-7?,2J /0/8*;2 1--3L0,68 ,6 .7-/1-/80 E \ b 1-8-3-M ~80 .39;* 9b79J8/9\ /--8b*8/8b*,,-\ -/9;0 6+/0// 0 -3L0,68 /;= 30/= Ú= { ~8-M /0/8*;* 1--3L0,68 O-12/ ~770./6 0 .3-0[b-7?,69 8-)16 .7-/1-/80 0;*K J8\ /--8b*8/8b*,,-\ 1--3L0,68 F (−c ; 0) \ F (c ; 0) \ M (x; y) \ 6 /--8,-I*,0* ò^ .30,0;6*8 b0L p p òò (x + c) + y + (x − c) + y = 2a. Á¸ 2

2

2

2

2

2

2

2

2

2

2

1

2

2

1

1

1

1

1

1

1

1

1

1

2

2

1

2

1

2

1 2

1 2

1 2

2

1

2

2

2

2

2

H*3*,-/9 -L0, 0[ 36L0167-b b .36b2J )6/8? 36b*,/8b6\ 0 [68*; b-[b-L9 Lb6 36[6 b 1b6L368\ -/b-+-L0;/9 -8 03360-,67?,-/80 = H-/7* ,*/7-|,N .3*K -+36[-b6,0M .-72)0; /7*L2J}** 236b,*,0*\ 9b79J}**/9 /7*L/8b0*; 236bK ,*,09 òò x y ò_ + = 1, a b :L* b = a − c = 2

2

2

2

2

2

2

y

(0, b) M (x, y)

d1 r1

d2

r2

F1 (−c, 0)

F2 (c, 0)

x

(a, 0)

(x = − ae )D1

D2 (x = ae )

Ç0/= Ú= K *832L,- 2+*L08?/9\ )ò_8- b/9169 8-)16 M (x; y) \1--3L0,68 1-8-3-M 2Lò^ b7*8b-39J8 236b,*,0J \ .30,6L7*|08 36//;6830b6*;-;2 ~770./2 = *M/8b08*7?,-\ 236b,*,0* ò_ ~1b0b67*,8,- /7*L2J}*;2 y = b − x = H-L/86b799 ~8- b36|*,0* L79 y b O-3;27 L79 36//8-9,09 -8 M L- O-K 12/-b |−M−F→| = p(x + c) + y 0 |−M−F→| = p(x − c) + y \ .-/7* .30b*L*,09 .-L-+,N )7*,-b .-72)0; −−→ −−→ c c òã |M F | = a + x, |M F | = a − x, a a -812L6 /7*L2*8\ )8- |−M−F→| + |−M−F→| = 2a \ 8- */8? M ∈ Φ = [6L6, b ,*1-8-3-M .39;-K 610; -+36[-;\ b/910M ~770./ ;-|*8 +8? 2:-7?,-M /0/8*;* 1--3L0,68 236b,*,0*; ò_= 8- 236b,*,0* ,6[b6*8/9 ¤= 0/8*;6 1--3L0,68\ b 1-8-3-M ~770./ 0;**8 236b,*,0* ò_\ ,6[b6*8/9 ¤¦ L79 ~8-:- ~770./6= Á¾ 2

2

1

2

2

2

1

1

2

2

2

2

2

b2 2 a2

ðÍµxñÍXW´Z = [ 236b,*,09 ò_ /7*L2*8\ )8- ª Ã¡ Φ Φ ¥ À« ªº ¦ ÃÑ e = e = e ¥ ¡ª¬= *M/8b08*7?,-\ b ~8-; /72)6* .636;*83 a 0 b 0[ 16,-,0)*/10N 236b,*,0M ~80N ~770./-b -16[K b6J8/9 .3-.-30-,67?,;0 */70 a = λa \ 8- c = a e = λa e = λc \ 6\ /7*L-b68*7?,-\ 0 b = λb = ¼/70 36/.-7-|08? 8*.*3? ~770./ Φ 0 Φ ,6 .7-/1-/80 861\ )8-+ 16,-,0)*/10* /0/8*; 1--3L0,68 2 ,0N /-b.670\ 8-,0\ -)*b0L,-\ -16|28/9 :-;-8*80),;0 / 1-~OO00*,8-; :-;-8*800 λ = ]= )*b0L,-\ L79 16|L-:- )0/76 e < 1 ;-|,- .-L-+368? Lb6 8610N 2:76 ) ò α, β ∈ (0, ) \ 8- +2L*8 b.-7,98?/9 /--8,-I*,0* \ .-~8-;2 b/910M ~7K 70./\ -.3*L*79*;M .3-0[b-7?,;0 [,6)*,09;0 .636;*83-b a 0 c \ ;-|,3*670[-b68? 161 .*3*/*)*,0* 1-,2/6 / .7-/1-/8?J= Æ= { )6/8,-/80\ -8/JL6 /7*L2*8\ )8- b/910M ~770./\ -870),M -8 -132|K ,-/80\ 0;**8 L03*1830/= H-/1-7?12 8-)16 M \ 2L-b7*8b-39J}69 /--8,-I*,09; òã\ .30,6L7*K |08 ~770./2 ò_\ 8- O-3;27 òã .3*L/86b79J8 /-+-M b36|*,09 L79 O-167?,N 36L02/-b 8-)10 M = 610; -+36[-;\ ; .-72)070 /7*L2J}0* b36|*,09 L79 O-167?,N 36L02/-b 8-)10 M (x; y) \ .30,6L7*|6}*M ~7K 70./2 Φ \ [6L6,,-;2 236b,*,0*; ò_ òa r = a + ex, r = a − ex. *.*3? ,*832L,- 2+*L08?/9\ )8- .39;* D 0 D \ 0;*J}0*\ /--8b*8K /8b*,,-\ 236b,*,09 x = − 0 x = \ 9b79J8/9 L03*1830/6;0 ~770./6= *M/8b08*7?,-\ 36//8-9,09 -8 8-)10 M (x; y) L- .39;N D 0 D 36b, dist(M, D ) = + x 0 dist(M, D ) = − x \ -812L6 /7*L2*8 r a ± ex r ò` = e. = = ±x dist(M, D ) dist(M, D ) b-M/8b- ~770./6\ b36|6*;-* /--8,-I*,09;0 òÚ 070 ò`\ 9b79*8/9 /b-M/8b-;\ -.3*L*79J}0; ~770./= -),**\ 0;**8 ;*/8- /7*L2J}** .3*L7-K |*,0* = TxwuxXWxZ ª¦ ¡ ¢ª ¡££ D ¤ F ¥ ¡ª»«£ ¡£¦ D ¥ ¡ ¢ª ¡» ÁÁ 1

1

2

2

2

2

1

2

2

1

1

1

1

2

π 2

1

2

1

a e

2

a e

1

1

a e

2

a e

2

1

1

2

a e

2

¤ e < 1 § ¨©ª » ¤ ¡¥ ª£ Ð Äº ££ ª ¤ F £À ª ¡£¦ D ¤ e ¥ £££ Ã¡ ÃÑ e § Ã F Ã ª ¢ É Ã© Ã¡¥ D ª¥ À«£ É F § ðÍwÍñÍ ÊrZ -16[68? ~8- .3*L7-|*,0* = óyÍêÍXWxZ H2/8? 36//8-,0* -8 8-)10 F L- .39;-M D 36b,- h = [ /0/8*; 236b,*,0M a − c = h, e a c

c =e a

-L,-[,6),- ,6N-L98/9 )0/76 0 = *.*3? ,6 .7-/1-/80 ;-|,- b+368? /0/8*;2 1--3L0,68\ b 1-8-3-M .39;69 D 0;**8 236b,*,0* x = − \ 6 8-)K 16 F 0;**8 1--3L0,68 (−c ; 0) = /86*8/9 [6.0/68? 236b,*,0* |−M−→F | = ) ò_ e · dist(M, D) 0 .-16[68?\ 8- -,- /-b.6L6*8 / 236b,*,0*; = ) 8;*80; ,*1-8-3* 8*3;0,\ 0/.-7?[2*;* .30 36//;-83*,00 ~770./6 0/7- a ,6[b6*8/9 ¬Ä¦ ¡À ~770./6Å )0/7- b ,6[b6*8/9 ¦ ¡À ~770./6Å )0/7- 2c ,6[b6*8/9 É £Å ,6)67- 16,-,0)*/1-M /0/8*; 1--3L0,68\ 9b79J}**/9 *,83-; /0;;*8K 300 ~770./6\ ,6[b6*8/9 Ñ ~770./6Å -/0 16,-,0)*/1-M /0/8*; 1--3L0,68\ 9b79J}0*/9 -/9;0 /0;;*8300 ~7K 70./6\ ,6[b6J8/9 £ ~770./6Å -/? 6+/0// 16,-,0)*/1-M /0/8*; 1--3L0,68 ,6[b6J8/9 ¬Ä¦ 070 É¦ À ~770./6Å -/? -3L0,68 16,-,0)*/1-M /0/8*; 1--3L0,68 ,6[b6J8/9 ¦ À ~770./6Å 8-)10 .*3*/*)*,09 ~770./6 / *:- -/9;0\ 0;*J}0* 1--3L0,68 (±a; 0) 0 (0; ±b) \ ,6[b6J8/9 Ä ~770./6 = àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*109 ãÅ âä \ :76b6 @@\ å]Å âòä \ :76b6 _\ åå\ ]\ Æ= ðÍwÍñW W S³TÍXxXW´á â]ä \ Æ`]\ Æ`ã\ Ú^\ Ú^Æ\ Ú^\ Ú\ ÚÆ\ ÚÆã\ ÚÆa\ Úò]\ ÚòÆ\ Ú_Ú\Ú_ò\ Ú_a\ Úã\ Úã]\ ÚãÆ\ Úã_\ ÚããÅ âÚä \ 8*; ò\ _= a e

ÁÈ

sSñÍÌ ±ø β = α Z { ~8-; /72)6* .7-/1-/8? π .63677*7?,6 -L,-M 0[ -+36[2J}0N 1-,2/6 0 .-~8-;2 .*3*/*16*8 b/* -+36[2J}0* 13-;* -L,-M= /2}*/8b0; /7*L2JK }0* .-/83-*,09 /;= 30/= Ú]= {.0I*; b 1-,2/ Φe /O*32 Σ \ 16/6J}2J/9 .7-/1-/80 π = H2/8? F · 8-)16 16/6,09 /O*3 Σ / .7-/1-/8?J π = O*36 Σ 16/6*8/9 1-,2/6 Φe .- -132|,-/80\ 7*|6}*M b .7-/1-/80\ .*3.*,L012793K ,-M -/0 1-,2/6= +-[,6)0; 216[6,,* -132|,-/8? 0 .7-/1-/8? )*3*[ ω 0 ρ /--8b*8/8b*,,- = +-[,6)0; 861|* )*3*[ D .39;2J\ .- 1-8-3-M .7-/1-/8? π .*3*/*16*8 .7-/1-/8? ρ =

Ç0/= Ú]= H2/8? M · .3-0[b-7?,69 8-)16 1-,0)*/1-:- /*)*,09 Φ \ A · 8-)16 .*K 3*/*)*,09 -+36[2J}*M SM / -132|,-/8?J ω \ C · -/,-b6,0* .*3.*,L012K ÁÔ

7936\ -.2}*,,-:- 0[ 8-)10 M ,6 .39;2J D \ 6 P · -/,-b6,0* .*3.*,L012K 7936\ -.2}*,,-:- 0[ 8-)10 M ,6 .7-/1-/8? ρ = H-/1-7?12 b/910* Lb* 16/68*7?,* 1 /O*3*\ .3-b*L*,,* 0[ -L,-M 0 8-M |* 8-)10\ 36b,\ 8- |−M−→F | = |−M−→A| \ 6 .-/1-7?12 α = β \ 8- .39;-2:-7?,* 83*2:-7?,010 M P A 0 M P C 36b,\ -812L6 /7*L2*8\ )8- |−M−→A| = |−M−→C| = 6K 10; -+36[-;\ 16|L69 8-)16 M 1-,0)*/1-:- /*)*,09 Φ b 36//;6830b6*;-; /72)6* 2L-b7*8b-39*8 /7*L2J}*;2 2/7-b0J −−→ −−→ _^ |M F | = |M C|. .7-/1-/80\ 2L-b7*8b-39J}0N 2/7-b0J _^\ °,-|*/8b- 8-)*1 *b170L-b-M ,6[b6*8/9 ¡¬¦= -),**\ /O-3;27032*; /7*L2J}** -.3*L*7*,0* = Ï³TxwxxXWxZ ¡ E ª ¡££ D ¤ F ¥ ¡ª»«£ ¡£¦ D § ¬¦ É F ª ª¦ D º ¢£ » Φ Ð ¤ M ¡¥ £ Ð £º ª ¤ F £À ª ¡£¦ D ¥ ¥ −−→ _ Φ = {M ∈ E : |M F | = dist(M, D)}. H2/8? 8-)16 M 7*|08 ,6 .636+-7* Φ = >0/7- r = |−M−→F | ,6[b6*8/9 Éº ª 8-)10 M = H2/8? d -[,6)6*8 36//8-9,0* -8 8-)10 M ∈ Φ L- L03*1830/= H-/1-7?12 L79 b/91-M .636+-7 -8,-I*,0* O-167?,-:- 36L02/6 ** 8-)10 M 1 36//8-K 9,0J -8 ~8-M 8-)10 L- L03*1830/ ,* [6b0/08 -8 b+-36 8-)10 M 0 36b,\ 8- .- -.3*L*7*,0J /)086*;\ )8- b/9169 .636+-76 0;**8 ÃÑº e = 1 = 8- -.3*L*7*,0* /-:76/2*8/9 0 / O-3;27-M ò\ .-/1-7?12 b 36//;6830b6*;-; /72)6* cos β = cos α = ,ÍXuXWñx¶yux STÍtXxXWx ³ÍTÍvuUZ Ç6//;-830; .636+-72 Φ \ -.3*L*7*,,2J 161 ;,-|*/8b- 8-)*1 _= H2/8? 36//8-9,0* -8 O-12/6 F .636+-7 Φ L- ** L03*1830/ D 36b,- p \ 8- */8? H ) dist(F, D) = p = 39;69\ .3-N-L9}69 *3*[ O-12/ F 0 .*3.*,L012793,69 L03*1830/*\ 9b79*8/9 -/?J /0;;*8300 .636+-7= {+*3*; .39;-2:-7?,2J /0/8*;2 1--3L0,68 ,6 .7-/1-/80 E \ b 1-8-3-M ~86 .39;69 9b79*8/9 -/?J 6+/0//\ ,6)67- 1--3L0,68 36b,-2L67*,- -8 O-12/6 0 L03*1830/\ 6 ,6K .36b7*,0* -/0 6+/0// /-b.6L6*8 / ,6.36b7*,0*; -8 L03*1830/ 1 O-12/2 ÁÙ 2

2

2

/;= 30/= ÚÆ= { ~8-M /0/8*;* 1--3L0,68 O-12/ .636+-7 0;**8 1--3L0,68 H F ( ; 0) \ 6 L03*1830/6 0;**8 236b,*,0* x = − = 2/8? M (x; y) · .3-0[K b-7?,69 8-)16 .636+-7= --8,-I*,0* _ .30,0;6*8 b0L r p p _] + y = x + . x− 2 2 {-[b-L9 -+* )6/80 ~8-:- 36b*,/8b6 b 1b6L368 0 .30b-L9 .-L-+,* )7*,\ .-72)0; /7*L2J}** 236b,*,0*\ ~1b0b67*,8,-* 236b,*,0J _] _Æ y = 2px. .636+-7= 0/8*;6 8- 236b,*,0* ,6[b6*8/9 ¤ 236b,*,0*; 1--3L0,68\ b 1-8-3-M .636+-76 0;**8 236b,*,0* _Æ\ ,6[b6*8/9 º ¤¦ L79 ~8-M .636+-7= p 1 2

p 2

2

2

2

y d

M (x, y) r

O

x

F (p/2, 0)

D (x = − p2 )

0/= ÚÆ= Ç ðÍµxñÍXW´Z = [ 236b,*,09 _Æ /7*L2*8\ )8- 7J+* Lb* .636+-7 Φ 0 Φ .-L-+K ,= *M/8b08*7?,-\ */70 36/.-7-|08? .636+-7 Φ 0 Φ ,6 .7-/1-/80 861\ )8-+ 16,-,0)*/10* /0/8*; 1--3L0,68 2 ,0N /-b.670\ 8- -,0 -16|28/9 :-;-8*80),;0 / 1-~OO00*,8-; :-;-8*800 λ = = 79 .3-b*310 ~8-:O6186 L-/868-),- 2/86,-b08?\ )8- .39;69 x = `t \ y = mt .*3*/*16*8 .6K 36+-7\ [6L6,,* 236b,*,09;0 y = 2p x 0 y = 2p x \ /--8b*8/8b*,,-\ b 8-)16N / 1--3L0,686;0 ( ; 2p `) 0 ( ; 2p `) = 1

1

2

p2 p1

2

2p1 `2 m

1 2p2 `2 m

1

ÈÝ

2

2

2

2

]= ¼/70 .7-/1-/8? π \ .*3*/*16J}2J 1-,2/ Φe .- .636+-7* Φ .*3*;*}68? .63677*7?,- /*+*\ 8- .30 ~8-;\ -)*b0L,-\ ;-|,- ,6M80 861-* ** .-7-|*K ,0*\ .30 1-8-3-; .636;*83 dist(F, D) = p .30;*8 7J+-* ,6.*3*L [6L6,,-* [,6)*,0* = H-~8-;2 7J+2J .636+-72\-.3*L*79*;2J .3-0[b-7?,; [,6)*,0K *; .636;*836 p \ ;-|,- 3*670[-b68? 161 .*3*/*)*,0* 1-,2/6 / .7-/1-/8?J= H-/1-7?12 .63677*7?,* .7-/1-/80\ -)*b0L,-\ :-;-8*80), / *,83-; :-K ;-8*800 b 7J+-M\ ,* .30,6L7*|6}*M 0; 8-)1*\ 0\ b )6/8,-/80\ / *,83-; :-;-8*800 b 8-)1* S \ 8- 0 .636+-7\ b/*16*;* 1-,2/-; Φe ,6 ~80N .7-/K 1-/89N :-;-8*80),\ 6 /7*L-b68*7?,- 0 .-L-+,= 8;*80; ,*1-8-3* 8*3;0,\ 0/.-7?[2*;* .30 36//;-83*,00 .636+-K 7 )0/7- p ,6[b6*8/9 É ¡ .636+-7Å -/? -3L0,68 16,-,0)*/1-M /0/8*; 1--3L0,68\ 9b79J}69/9 -/?J /0;K ;*8300 .636+-7\ ,6[b6J8/9 À .636+-7Å ,6)67- 16,-,0)*/1-M /0/8*; 1--3L0,68 (0; 0) \ 9b79J}**/9 8-)1-M .*K 3*/*)*,09 .636+-7 / ** -/?J\ ,6[b6*8/9 Ä¦ .636+-7= àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*109 ãÅ âä \ :76b6 @@\ åÅ âòä \ :76b6 _\ åå\ ]\ Æ= ðÍwÍñW W S³TÍXxXW´á â]ä \ _]ã\ _]a\ _]`\ _Æ^\ _Æ\ _Æò\ _Æ_\ _ÆãÅ âÚä \ 8*;6 a= sSñÍÌ Êø β < α Z { ~8-; /72)6* .7-/1-/8? π .*3*/*16*8 1-,2/ Φe .- -+* /8-3-, -8 *:b*3I0,= P61 0 b .*3b-; /72)6*\ b.0I*; b 1-,2/ Lb* /O*3 Σ 0 Σ \ 16/6J}0*/9 .7-/1-/80 π /;= 30/= ÚÚ\ 0 bb*L*; -+-[,6)*,09 F 0 F L79 8-)*1 16/6,09 /O*3 Σ 0 Σ / .7-/1-/8?J π \ ω 0 ω L79 -132|,-/8*M\ .- 1-8-3; /O*3 Σ 0 Σ 16/6J8/9 1-,2/6 Φe \ ρ 0 ρ L79 .7-/1-/8*M\ b 1-8-3N 36/.-7-|*, -132|,-/80 ω 0 ω \ D 0 D L79 .39;N\ .1-8-3; .7-/1-/8? π .*3*/*16*8 .7-/1-/80 ρ 0 ρ /--8b*8/8b*,,- = 8-)12 M 1-,0)*/1-:- /*)*,09 Φ \ 36/.-7-K Ç6//;-830; .3-0[b-7?,2J |*,,2J b 8-M |* )6/80 1-,2/6 Φe \ )8- 0 -132|,-/8? ω = +-[,6)0; )*3*[ ) ) ) A 0 A 8- 10 .*3*/* *,09 -+36[2J}*M SM / -132|,-/89;0 ω 0 ω \ *3*[ ) C 0 C -/,-b6,09 .*3.*,L012793-b\ -.2}*,,N 0[ 8- 10 M ,6 .39;* ) ) D 0 D \ 6 *3*[ P 0 P -/,-b6,09 .*3.*,L012793-b\ -.2}*,,N 0[ 8- 10 ÈÜ 1

1

1

1

2

1

1

1

2

1

2

2

2

2

1

2

2

2

1

2

1

2

1

2

1

1

2

2

2

M

,6 .7-/1-/80 ρ 0 ρ = 1

2

Ç0/= ÚÚ = H-/1-7?12 b/910* Lb* 16/68*7?,* 1 /O*3*\ .3-b*L*,,* 0[ -L,-M 0 8-M |* 8-)10\ 36b,\ 8- |−M−F→| = |−M−−A→| \ 6 |−M−F→| = |−M−−A→| = 8/JL6 /7*L2*8\ )81

1

2

2

−−→ −−→ −−−→ −−−→ −−−→ |M F1 | − |M F2 | = |M A1 | − |M A2 | = |A1 A2 |.

79 .3-0[b-7?,-M 8-)10 M ∈ Φ \ 36/.-7-|*,,-M b 8-M |* )6/80 1-,2/6 Φe \ )8- 0 -132|,-/8? ω \ 6,67-:0),- .-72)0; 2

−−→ −−→ −−−→ |M F1 | − |M F2 | = |A1 A2 |. −−−→ |A1 A2 | ω1 ω2

.3*L/86b79*8 /-+-M L70,2 -83*[16 -+36[2J}*M 61 161 b*70)0,6 0 \ 0 ,* [6b0/08 -8 bK SM 1-,2/6\ 7*|6}*:- ;*|L2 -132|,-/89;0 Èï

+-36 8-)10 M ∈ Φ \ 8- 16|L69 8-)16 M 1-,0)*/1-:- /*)*,09 Φ b 36//;68K 30b6*;-; /72)6* 2L-b7*8b-39*8 /7*L2J}*;2 2/7-b0J −−→ −−→ _Ú |M F | − |M F | = const. .7-/1-/80\ 2L-b7*8b-39J}0N 2/7-b0J _Ú\ °,-|*/8b- 8-)*1 *b170L-b-M ,6[b6*8/9 ©¡¬¦= -),**\ /O-3;27032*; /7*L2J}** -.3*L*7*,0* = Ï³TxwxxXWxZ ¡ E ª ª ¤ F F ¥ £º »ª 2c § .¡¬¦ É F F ¢º £ » Φ Ð ¤ M ¡¥ ¬À£ ¤ ¢º £¦ Ð ª F F ¡£ ¤ 2a < 2c ¥ ¥ −−→ −−→ _ò Φ = {M ∈ E : |M F | − |M F | = 2a}. 1

2

2

1

2

1

1

2

1

2

2

8-),-/8?J L- .-7-|*,09 ,6 .7-/K 1-/80 :0.*3+-76 -.3*L*79*8/9 .636;*8K 36;0 2a 0 2c = 8,-I*,0* ~80N .636K ;*83-b e=

2

1

2

1

2

1

2

F1

O1 S

A1

α A2 F2

2

1 2

1

β

c >1 a

,6[b6*8/9 ÃÑ :0.*3K +-7= 79 :0.*3+-7\9b79J}*M/9 1-,0)*K /10; /*)*,0*;\.636;*83 2a 36b*, L70,* -83*[16 A A b,283*,,*M -+}*M 16/6K 8*7?,-M .39;-M 1 /O*36; Σ 0 Σ \ 6 .636;*83 2c 36b*, L70,* -83*[16 F F b,*I,*M -+}*M 16/68*7?,-M .39;-M 1 ~80; /O*36;= H-/1-7?12 /;= 30/= Úò −−−→ −−−→ −−→ |A A | = |O O | cos α \ 6 |F F | = −−−→ |O O | cos β \ 8cos β __ . e= 1

2

O2

1 2

cos α

È

Ç0/= Úò=

0/76 r = |−M−F→| 0 r = |−M−F→| ,6[b6J8/9 É ª > 8-)10 M = [ 36//;-83*,09 .39;-2:-7?,N 83*2:-7?,01-b M P A 0 M P C L79 :0.*3+-7 Φ \ 9b79J}*M/9 1-,0)*/10; /*)*,0*; 30/= Ú_\ .-72)6*;\ 161 0 b /72)6* ~770./6\ )8- |−M−P→| = |−M−−C→| cos β \ |−M−P→| = |−M−−A→| cos α = 8/JL6 /7*L2*8\ )8−−→ −−−→ |M F | |M A | cos β _ã = e. −−−→ = −−−→ = 1

1

2

2

2

2

2

|M C2 |

2

2

2

2

2

2

2

|M C2 |

cos α

Ç0/= Ú_= Ç6//;6830b69 .39;-2:-7?,* 83*2:-7?,010 M P A 0 M P C \ .30L*; 1 È¸ 1

1

1

1

6,67-:0),-;2 /--8,-I*,0J _a

−−−→ −−→ |M A1 | |M F1 | −−−→ = −−−→ = e. |M C1 | |M C1 |

H39;* D 0 D ,6[b6J8/9 ª :0.*3+-7 Φ = +-[,6)0; 36/K /8-9,09 -8 8-)10 M :0.*3+-7 L- L03*1830/ )*3*[ d = |−M−−C→| \ d = −−−→ _ã _ |M C | = [ /--8,-I*,0M 0 a b8*16*8 /7*L2J}** /b-M/8b- :0.*3+-K 7= É© ª ¤ M ©¡¬ £À ¿Ä Ã¦ ¤ ª À«¦ ª ¢ ¬ ¤º M ÃÑ ©¡¬ r r _` = = e. d d ðÍµxñÍXWxZ {I*.30b*L*,,* 36//2|L*,09 -8,-/98/9 1 :0.*3+-7*\ 9bK 79J}*M/9 1-,0)*/10; /*)*,0*;= 0|* +2L*8 216[6,-\ 161 ,6N-L98/9 L0K 3*1830/ .3-0[b-7?,-M :0.*3+-7\ -.3*L*7*,,-M 2/7-b0*; _ò= ,ÍXuXWñx¶yux STÍtXxXWx VW³xTvuUZ Ç6//;-830; :0.*3+-72 Φ)\ -.3*L*7*,,2J 161 ;,-|*/8b- 8-)*1 _ò= [ ~8-:- -.3*L*7*,09 /7*L2*8\ 8- .39;69 F F 0 .*3.*,L012793 1 F F \ L*K 79}0M -83*[-1 F F .-.-76;\ 9b79J8/9 -/9;0 /0;;*8300 :0.*3+-7= {K +*3*; .39;-2:-7?,2J /0/8*;2 1--3L0,68 ,6 .7-/1-/80 E \ b 1-8-3-M ~80 .39;* 9b79J8/9\ /--8b*8/8b*,,-\ -/9;0 6+/0// 0 -3L0,68 /;= 30/= Úò= { ~8-M /0/8*;* 1--3L0,68 O-12/ ~770./6 0 .3-0[b-7?,69 8-)16 .7-/1-/80 0;*J8\ /--8b*8/8b*,,-\ 1--3L0,68 F (−c ; 0) \ F (c ; 0) \ M (x; y) \ 6 /--8,-K I*,0* _ò .30,0;6*8 b0L p p ã^ (x + c) + y − (x − c) + y = ±2a. H*3*,-/9 b8-3-M 36L0167 b .36b2J )6/8? 36b*,/8b6\ 0 [68*; b-[b-L9 Lb6 36[6 b 1b6L368\ -/b-+-|L6*;/9 -8 03360-,67?,-/80= H-/7* ,*/7-|K ,N .3*-+36[-b6,0M .-72)0; /7*L2J}** 236b,*,0*\9b79J}**/9 /7*L/8b0K *; 236b,*,09 ã^ y x ã − = 1, a b :L* b = c − a = È¾ 2

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D2

D1 d2

d1 M

r2 F2

r1 F1

O

Ç0/= Úã= K *832L,- 2+*L08?/9\ )8-ã b/9169 8-)16 M (x; y) \ 1--3L0,68 1-8-3-M 2L_ò b7*8b-39J8 236b,*,0J \.30,6L7*|08 36//;6830b6*;-M :0.*3+-7* = *M/8b08*7?,-\ 236b,*,0* ã ~1b0b67*,8,- /7*L2J}*;2 y = x − b = H-L/86b799 ~8- b36|*,0* L79 y b O-3;27 L79 36//8-9,09 -8 M L- O-K 12/-b |−M−F→| = p(x + c) + y 0 |−M−F→| = p(x − c) + y \ .-/7* .30b*L*,09 .-L-+,N )7*,-b .-72)0;\ )8- L79 8-)*1 :0.*3+-7 / 6+/0//-M x 6 −a −−→ −−→ c c ã] |M F | = −a − x, |M F | = a − x, a 6 L79 8-)*1 :0.*3+-7 / 6+/0//-Ma x > a −−→ −−→ c c ãÆ |M F | = a + x, |M F | = −a + x, a a − − → − − → ) -812L6 /7*L2*8\ 8- |M F | − |M F | = ±2a \ 8- */8? M ∈ Φ = 610; -+36[-;\ b/9169 :0.*3+-76 ;-|*8 ã+8? [6L6,6 b ,*1-8-3-M .39K ;-2:-7?,-M /0/8*;* 1--3L0,68 236b,*,0*; = 8- 236b,*,0* ,6[b6*8/9 ¤= 0/8*;6 1--3L0,68\ b 1-8-3-M ~770./ 0;**8 236b,*,0* ã\ ,6[b6*8/9 ¤¦ L79 ~8-M :0.*3+-7= ðÍµxñÍXW´Z = [ 236b,*,09 ã /7*L2*8\ )8- ª ©¡¬ Φ Φ ¥ À« ªº ¦ ÃÑ e = e = e ¥ ¡ª¬= *M/8b08*7?,-\ b ~8-; /72)6* .636;*83 a 0 b 0[ 16,-,0)*/10N 236b,*,0M ~80N ~770./-b -16[K b6J8/9 .3-.-30-,67?,;0 */70 a = λa \ 8- c = a e = λa e = λc \ 6\ ÈÁ b2 2 a2

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/7*L-b68*7?,-\ 0 b = λb = ¼/70 36/.-7-|08? 8*.*3? :0.*3+-7 Φ 0 Φ ,6 .7-/1-/80 861\ )8-+ 16,-,0)*/10* /0/8*; 1--3L0,68 2 ,0N /-b.670\ 8-,0\ -)*b0L,-\ -16|28/9 :-;-8*80),;0 / 1-~OO00*,8-; :-;-8*800 λ = ]= )*b0L,-\ L79 16|L-:- )0/76 e > 1 ;-|,- .-L-+368? Lb6 8610N 2:76 ) __ α, β ∈ (0, ) \ 8- +2L*8 b.-7,98?/9 /--8,-I*,0* \ .-~8-;2 b/9169 :0.*3+-76\ -.3*L*79*;69 .3-0[b-7?,;0 [,6)*,09;0 .636;*83-b a 0 c \ ;-|*8 +8? 3*670[-b6,6 161 .*3*/*)*,0* ,*1-8-3-:- 1-,2/6 / .7-/1-/8?J= Æ= { )6/8,-/80\ -8/JL6 /7*L2*8\ )8- b/9169 :0.*3+-76 0;**8 L03*1830/= Ú = Q0.*3+-76 /-/8-08 0[ Lb2N ¦\ -.3*L*79*;N b 16,-,0)*/1-M /0K /8*;* 1--3L0,68 /--8,-I*,09;0 x 6 −a 0 x > a /--8b*8/8b*,,- = H- -8,-K I*,0J 1 16,-,0)*/1-M /0/8*;* 1--3L0,68 .*3b69 0[ ,0N ,6[b6*8/9 ¦ À\ 6 b8-369 ¡¦ À= H-/1-7?12 8-)16 M \2L-b7*8b-39J}69 /--8,-I*,09; ã] 070 ãÆ\.30K ,6L7*|08 :0.*3+-7* ã\ 8- O-3;27 ã] .3*L/86b79J8 /-+-M b36|*,09 L79 O-167?,N 36L02/-b 8-)10 M (x; y) \ .30,6L7*|6}*M 7*b-M b*8b0 :0K .*3+-7 Φ \ [6L6,,-M 236b,*,0*; ã ãÚ r = −a − ex, r = a − ex, 6 O-3;27 ãÆ .3*L/86b79J8 /-+-M b36|*,09 L79 O-167?,N 36L02/-b 8-)10 M (x; y) \ .30,6L7*|6}*M .36b-M b*8b0 ~8-M :0.*3+-7 ãò r = a + ex, r = −a + ex. )*.*3? ,*832L,- 2+*L08?/9 6,67-:0),- /72)6J ~770./6\ ,- b L6,,-;) /72 6* ,*-+N-L0;- L79 16|L-M L03*1830/ -8L*7?,- 36//;6830b68? 8- K 10\ .30,6L7*|6}0* 7*b-M 0 .36b-M b*8b9; :0.*3+-7\ )8- .39;* D 0 D \ 0;*J}0*\ /--8b*8/8b*,,-\ 236b,*,09 x = − 0 x = \ 9b79J8/9 L03*1K 830/6;0 :0.*3+-7\ 8- */8? )8- L79 7J+-M 8-)10 :0.*3+-7 b.-7,9J8/9 /--8,-I*,09 _`= b-M/8b- :0.*3+-7\ b36|6*;-* /--8,-I*,09;0 _`\ 9b79*8/9 /b-MK /8b-;\ -.3*L*79J}0; :0.*3+-72= -),**\ 0;**8 ;*/8- /7*L2J}** .3*L7-K |*,0* = TxwuxXWxZ ª¦ ¡ ¢ª ¡££ D ¤ F ¥ ¡ª»«£ ¡£¦ D ¥ ¡ ¢ª ¡» ÈÈ 2

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π 2

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a e

a e

¤ e > 1 § ¨©ª » ¤ ¡¥ ª£ Ð Äº ££ ª ¤ F £À ª ¡£¦ D ¤ e ¥ £££ ©¡¬¦ ÃÑ e § Ã F Ã ª ¢ É Ã¦ ©¡¬¥ D ª¥ À«£ É F§ ðÍwÍñÍ Ê±Z -16[68? ~8- .3*L7-|*,0*\ b-/.-7?[-b6bI0/? 216[6,0*; 1 [6L6)* Æ /;= /= _ã= 8;*80; ,*1-8-3* 8*3;0,\ 0/.-7?[2*;* .30 36//;-83*,00 :0.*3+-K 7 )0/7- a ,6[b6*8/9 ª¦¦ ¡À :0.*3+-7Å )0/7- b ,6[b6*8/9 ¦ ¡À :0.*3+-7Å )0/7- 2c ,6[b6*8/9 É £Å ,6)67- 16,-,0)*/1-M /0/8*; 1--3L0,68\ 9b79J}**/9 *,83-; /0;;*8K 300 :0.*3+-7\ ,6[b6*8/9 Ñ :0.*3+-7Å -/0 16,-,0)*/1-M /0/8*; 1--3L0,68\ 9b79J}0*/9 -/9;0 /0;;*8300 :0K .*3+-7\ ,6[b6J8/9 £ :0.*3+-7Å -/? 6+/0// 16,-,0)*/1-M /0/8*; 1--3L0,68 ,6[b6J8/9 ª¦º ¦ 070 É¦ À :0.*3+-7Å -/? -3L0,68 16,-,0)*/1-M /0/8*; 1--3L0,68 ,6[b6J8/9 ¦ À :0.*3+-7Å 8-)10 .*3*/*)*,09 :0.*3+-7 / ** L*M/8b08*7?,-M -/?J\ 0;*J}0* 1--3K L0,68 (±a; 0) \ ,6[b6J8/9 Ä :0.*3+-7= ¶Wµ³zuzU VW³xTvuUZ Ç6//;-830; .7-/1-/8? π \ .3-N-L9}2J )*3*[ b*3I0,2 S 1-,2/6 Φe 0 .6K 3677*7?,2J .7-/1-/80 π \ b 1-8-3-M 7*|08 :0.*3+-76 Φ = Φe ∩ π = 86 .7-/K 1-/8? π .*3*/*16*8 1-,2/ Φe .- Lb2; *:- -+36[2J}0; ` 0 ` \ 1-8-3*\ -)*b0L,-\ ,* .*3*/*16J8 :0.*3+-72= H3-b*L*; 8*.*3? )*3*[ -+36[2J}0* ` 0 ` 16/68*7?,* .7-/1-/80 1 1-,2/2 Φe = +-[,6)0; 0N\ /--8b*8/8b*,,-\ )*K 3*[ γ 0 γ = H39;* ` = γ ∩ π 0 ` = γ ∩ π \ .- 1-8-3; ~80 .7-/1-/80 .*3*/*16J8 .7-/1-/8? π \ -)*b0L,-\ 7*|68 b .7-/1-/80 π 0 ,* 0;*J8 -+}0N 8-)*1 / 1-,2/-; .-/1-7?12 1-,2/ .*3*/*16*8/9 / γ 0 γ .- .39;; ` k` 0 ) ` k` = 7*L-b68*7?,-\ .39;* ` 0 ` ,* 0;*J8 -+}0N 8- *1 / :0.*3+-7-M e ∩ π = 80 Lb* .39;* ,6[b6J8/9 ¡ :0.*3+-7 = Φ=Φ ÈÔ 0

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{ 16,-,0)*/1-M L79 :0.*3+-7 /0/8*;* 1--3L0,68 6/0;.8-8 :0.*3+-K 7 0;*J8 236b,*,09 0 bx + ay = 0. bx − ay = 0 *832L,- 2+*L08?/9\) )8- .30 2L67*,00 8-)10 :0.*3+-7) -8 ** b*3I0, 36//8-9,0* -8 ~8-M 8- 10 L- -L,-M 0[ 6/0;.8-8 ,*-:36,0 *,,- 2+b6*8= 67:*+360)*/1-M 8-)10 [3*,09 6/0;.8-8 :0.*3+-7 +2L28 36//;-83*, b /7*L2J}*; .636:36O* = ðÍµxñÍXWxZ Ç6//;-830; ,*1-8-32J -+36[2J}2J ` 1-,2/6 Φe \ .*3*/*16K J}2J 1-,0)*/1-* /*)*,0* Φ 1-8-3-* ;-|*8 +8? 161 :0.*3+-7-M 861 0 ~770./-; 070 .636+-7-M 0 .3-b*L*; )*3*[ ,** 16/68*7?,2J .7-/1-/8? γ 1 1-,2/2 Φe = 86 16/68*7?,69 .7-/1-/8? .*3*/*)*8 .7-/1-/8? π .- .39;-M ) ) { ` = γ ∩ π \ 16/68*7?,-M 1 1-,0 */1-;2 /* *,0J Φ = ~8-; /;/7*\ 6/0;.K 8-8 :0.*3+-7 .3*L/86b79J8 /-+-M 16/68*7?,* 1 :0.*3+-7* b ** 4+*/1-K ,*),-2L67*,,N< 8-)16N = -),M /;/7 ~80; .-,9809; ;-|,- .30L68? b 36;16N 861 ,6[b6*;-M .3-*180b,-M :*-;*8300= 5/0;.8-8 :0.*3+-7 ;-|,- 861|* 36//;6830b68? 161 .3*L*7?,* .-K 7-|*,09 16/68*7?,N .39;N 1 :0.*3+-7* .30 ,*-:36,0)*,,-; 2L67*,00 8-)10 16/6,09 -8 b*3I0, :0.*3+-7= àxyuµxXwSxµÍ´ WzxTÍzSTÍá âãä \ 7*109 ãÅ âä \ :76b6 @@\ åÆÅ âòä \ :76b6 _\ åå\ ]\ Æ= ðÍwÍñW W S³TÍXxXW´á â]ä \ òÆ\ òÆ]\ òÆÆ\ òÆÚ\ òÆò\ òÆ_\ òÆa\ òÆ`\ òÚ^\ òÚ\ òÚ]Å âÚä \ 8*;6 ã= c/ 0plegij 1nnpmol p pmei2hnj ' H2/8? Φ · ~770./ 070 :0.*3+-76\ [6L6,,*\ /--8b*8/8b*,,-\ 16,-,0)*/10K ;0 236b,*,09;0 x y 070 x − y = 1, ã_ + =1 a b a b 6 ` · .39;69 / ,6.36b79J}0; b*18-3-; u = {l; m} \ .3-N-L9}69 )*3*[ 8-)12 M (x ; y ) \ [6L6,,69 .636;*830)*/10;0 236b,*,09;0 ãã x = x + lt, y = y + mt. ÈÙ 0

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H3*L.-7-|0;\ )8- .39;69 ` .*3*/*16*8 130b2J Φ = 79 ,6N-|L*,09 8-)*1 .*3*/*)*,09 ` / Φ .-L/86b0; 236b,*,09 ãã b ã_= Â2L*; 36//;6830b68? /36[2 -+* 130b* · 0 ~770./\ 0 :0.*3+-72= H30 ~8-; b- b/*N b36|*,09N\ :L* /8-98 Lb-M,* [,610 ± 070 ∓ \ b*3N,0M [,61 +2L*8 -8,-/08?/9 1 8-;2 /72)6J\1-:L6 36//;6830b6*8/9 ~770./\6 ,0|,0M 1 8-;2 /72)6J\1-:L6 36/K /;6830b6*8/9 :0.*3+-76= :0.*3+-7= 861\ 8-)10 .39;-M ` \ .30,6L7*|6K }0* 130b-M Φ \ -.3*L*79J8/9 [,6)*,09;0 .636;*836 t \ 2L-b7*8b-39J}0;0 236b,*,0J (x + lt) (y + mt) ãa ± = 1. a b ã 6/13b69 b /1-+10 0 /-+0369 .-L-+,* )7*,\ .-72)0; 236b,*,00 Ç a 236b,*,0* l lx m my y x ã` t ± ± ± − 1 = 0. + 2t + a b a b a b (36b,*,0* ã` · ~8- 1b6L368,-* 236b,*,0* -8,-/08*7?,- t \ [6 0/17J)*K ,0*; 8-:- /72)69\ 1-:L6 Φ · :0.*3+-76 0 l m a^ − = 0. a b Ï³TxwxxXWxZ è£¦ ¦ u = {l, m} ¥ ª£À«¦ ÄÀ (80) ¥ ¢£ ¡¤© ¡º £ ª£ ©¡¬ − = 1 ¥ ¡££ ` ¡£À« ¥ ª£À« ÄÀ (80) ¥ ¢£ ¡£¦ ¡º ¤© ¡£ ª£ Ã¦ ©¡¬§ [ a^ /7*L2*8\ )8- :0.*3+-76 0;**8 Lb6 6/0;.8-80)*/10N ,6.36b7*,09\ 0 ~80 ,6.36b7*,09 [6L6J8/9 b*18-36;0 {a; b} 0 {a; −b} = 610* ,6.36b7*,09 0;*J8 6/0;.8-8 :0.*3+-7 bx ± ay = 0 · .39;* ,* 0;*J}0* / :0.*3+-K 7-M -+}0N 8-)*1 236b,*,0* ã` b ~8-; /72)6* .3*b36}6*8/9 b .3-80b-3*K )0*= H39;69 6/0;.8-80)*/1-:- ,6.36b7*,09\ ,* 9b79J}69/9 6/0;.8-8-M\ .*3*/*16*8 :0.*3+-72 b -L,-M 8-)1* b ~8-; /72)6* ã` · 236b,*,0* .*3b-M /8*.*,0= { L67?,*MI*; .3*L.-76:6*;\ )8- ,6.36b7*,0* 36//;6830b6*;-M .39;-M ) ) H ã ` ,* 9b79*8/9 6/0;.8-80 */10; 0 8- ` .*3*/*16*8 Φ = -/1-7?12 ` · 2

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x2 a2

0

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ÔÝ

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1b6L368,-* 236b,*,0* -8,-/08*7?,- t \ 8- ~8- 236b,*,0* 0;**8 Lb6 b*}*K /8b*,,N 3*I*,09 t 0 t = H-L/86b799 ~80 3*I*,09 b ãã\ ,6ML*; Lb* -+K }0* 8-)10 M (x = x + lt ; y = y + mt ) 0 M (x = x + lt ; y = y + mt ) 130b-M Φ 0 .39;-M ` = Ï³TxwxxXWxZ ¿¢ M M ¡£¦ ` ¢£ Ð ª¦ ¦ Φ § 3 ª¦ ¢À » À ¡£À ` § -)16 A \ 9b79J}69/9 /*3*L0,-M N-3L M M \ /--8b*8/8b2*8 [,6)*,0J .636;*836 t \ 36b,-;2 t = (t + t )/2 = 1

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u

M1

D(u)

Ç0/= Úa= ) { )6/8,-/80\ ; ;-|*; M (x ; y ) · ~8- .3-0[b-7?,69 8- 16 .39;-M ` = .-7-|08? M = A = -:L6 t = 0 0\ /7*L-b68*7?,-\ t + t = 0 = H- 8*-3*;* {0*86 b ~8-; /72)6* 1-~OO00*,8 .30 t b 1b6L368,-; 236b,*,00 (79) 36b*, ,27J lx my a ± = 0. a b 861\ 1--3L0,68 /*3*L0, N-3L M M 2L-b7*8b-39J8 236b,*,0J a = H-/1-7?12 M M · ~8- .3-0[b-7?,69 N-3L6\ 0;*J}69 L6,,-* ,*6/0;.K 8-80)*/1-* ,6.36b7*,0* u = {l; m} ; -+-[,6)6*; ,6.36b7*,09 .3*LK /86b79J}0;0 0N b*18-36;0\ 8- ;,-|*/8b- /*3*L0, b/*N N-3L\ 0;*J}0N ,6.36b7*,0* u = {l; m} \ 2L-b7*8b-39*8 236b,*,0J lx my a] ± = 0. a b (36b,*,0*; a] [6L6*8/9 ,*1-8-369 .39;69\ .3-N-L9}69 )*3*[ *,83 130K b-M Φ = Ï³TxwxxXWxZ ££ D(u) ¥ À«£ (82) ¥ ¢£ ªº ¦ Φ ¥ ¡£» Ð ª ª© ¡¤© º ÔÜ 0

0

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A 2

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¡£ u = {l; m} ¥ ¥ ¡¥ ª¥ ¡£» ¡º ¤ ¡À u = {l; m} § [ 236b,*,09 a] /7*L2*8\ )8- L06;*83 D(u) 0;**8 ,6.36b79J}0M b*1K 8-3 m l aÆ v = {l ; m } = ∓ ; . b a Ó*:1- .3-b*308?\ )8- ,6.36b7*,0*\ [6L6b6*;-* b*18-3-; v \ ,* 9b79*8/9 6/0;.8-80)*/10;= 6ML*; L06;*83 D(v) \ /-.39|*,,M ,6.36b7*,0J v = H-L/86b799 b 236b,*,0* a] l 0 m b;*/8- l 0 m \ .-72)6*; 0

0

0

l 0 x m0 y ± 2 =0 a2 b

⇐⇒

∓

2

2

0

mx ly ± =0 a2 b2 a2 b2

⇐⇒

−mx + ly = 0.

-+36[-;\ ,6.36b79J}0; b*18-3-; L06;*836 D(v) 9b79*8/9 .*3b-K 610; ,6)67?,- [6L6,,M ,6.36b79J}0M b*18-3 N-3L u = {l; m} = Ï³TxwxxXWxZö D(u) D(v) ¢À£ ¡£» ªº ¦ Φ ¥ Ð ¡£ ( ¡£À« ) u = {l; m} v = {l ; m } ¢À£ ¡£» ¦ Φ § [ /--8,-I*,0M aÆ /7*L2*8\ )8- ,6.36b7*,09 u = {l; m} 0 v = {l ; m } /-.39|*, -8,-/08*7?,- 130b-M Φ 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 l ma ll mm aÚ =∓ ⇐⇒ ± = 0. m lb a b {/9169 .39;69 ,*6/0;.8-80 )*/1-:- ,6.36b7*,09\ .3-N-L9}69 )*3*[ *,83 130b-M Φ \ 9b79*8/9 L06;*83-;\ /-.39|*,,; ,*1-8-3-;2 ,6.36b7*,0J ~8- ,6.36b7*,0* ;-|,- ,6M80 0[ 236b,*,09 aÚ= Ï³TxwxxXWxZ . ¡£ ¦ Φ ¢£ ¡ º ª ¡£» ¡¦ u = {l; m} v = {l ; m ª} ¥ Φ © ª©§ 79 ,6N-|L*,09 :76b,N ,6.36b7*,0M 130b-M Φ ,6L- ,6M80 .632 ,6K .36b7*,0M u = {l; m} 0 v = {l ; m } \ 2L-b7*8b-39J}0N /0/8*;* 236b,*,0M mm ll aò ± = 0. ll + mm = 0, b [ .*3b-:- 236b,*,09 aò .-72)6*; lla= −mm H-L/86b799 ~8- /--8,-K = I*,0* b- b8-3-* 236b,*,0* aò\ .-72)6*; 1 1 a_ ∓ = 0. b a Ôï 0

0

0

0

2

0

0

2

2

0

0

0

0

2

0

0

0

0

0

2

0

2

2

2

0

0

[ 236b,*,09 a_ /7*L2*8\ )8-\ [6 0/17J)*,0*; /72)69\ 1-:L6 Φ · -132|K ,-/8? 0 − = 0 \ b- b/*N -/867?,N /72)69N 70+- m = 0 0 l = 0 \ 70+m = 0 0 l = 0 = 610; -+36[-;\ *L0,/8b*,,-M .63-M :76b,N ,6.36b7*,0M ~770./-b ,* b3-|L6J}0N/9 b -132|,-/8? 0 :0.*3+-7 9b79J8/9 ,6.36bK 7*,09 1--3L0,68,N -/*M 16,-,0)*/1-M /0/8*; 1--3L0,68 -/*M /0;;*8K 300= >8- 16/6*8/9 -132|,-/80\ 8- 7J+69 .636 b[60;,- -38-:-,67?,N ,6.36b7*,0M 9b79*8/9 .63-M :76b,N ,6.36b7*,0M= ,Í¶ÍzxëXUx y -W³¶S W VW³xTvuxZ H2/8? M (x ; y ) ∈ Φ = 06;*83 D 130b-M Φ \.3-N-L9}0M )*3*[ M (x ; y ) \ 0;**8 236b,*,09 x = x t \ y = y t ,6.-;,0;\ )8- L06;*83 .3-N-L08 )*3*[ *,83 O(0; 0)= H39;69 ` Φ \ .3-N-L9}69 )*3*[ M 0 0;*J}69 ,6.36b7*K ,0*\ /-.39|*,,-* ,6.36b7*,0J L06;*836 D \ .*3*/*16*8 130b2J Φ b Lb2N 8-)16N M 0 M \ 0 8-)16 M 9b79J}69/9 /*3*L0,-M -83*[16 M M \ 861|* .30,6L7*|08 130b-M Φ = H30 ~8-; t \ t 0 t = (t + t )/2 · 3*I*,09 1b6LK 368,-:- 236b,*,09 ã`= 8- b-[;-|,- 8-7?1- b 8-; /72)6*\ 1-:L6 t = t = Ï³TxwxxXWxZ Þ (79) ª ¡ªÀ«Ð Ä£ ª ª t = t ¥ ©£¥ ¤ ¡££ ` ¡ À Φ Ð ¡ Àº «Ð ¤Ð§ ¨¤ M (t ) = M (t ) ¢À ¡ Ã ª¦¦ ¤¦ ¡¤£ ` Φ § Ï³TxwxxXWxZ ££ ` Φ ¥ ¡Ðª£«£ ¤¢ ¤ M ∈ Φ º À«£ ¡¥ ¡£» ¡À ª D ¥ ¡Ðª£«© ¤¢ Ã ¤¥ ¢£ ¦ Φ ¤ M § H39;69 ` 9b79*8/9 16/68*7?,-M 1 Φ b 8-)1* M 8-:L6 0 8-7?1- 8-:L6\ 1-:L6 8-)16 M 9b79*8/9 Lb-M,-M 8-)1-M .*3*/*)*,09 ` 0 Φ = 6.36b79J}0; b*18-3-; L06;*836 D \ .3-N-L9}*:- )*3*[ 8-)12 M \9bK 79*8/9 b*18-3 u{x ; y } = 6.36b79J}0M b*18-3 v{l; m} 16/68*7?,-M ` Φ /-.39|*, b*18-32 u{x ; y } = [ 2/7-b09 /-.39|*,,-/80 aÚ 0;**; 1 b2 0

1 a2

0

0

0

0

0

0

0

2

0

1

1

2

0

1

2

2

1

1

0

0

M0

1

0

2

2

1

1

2

2

0

M0

0

0

0

0

0

0

M0

0

0

lx0 my0 ± 2 = 0, a2 b

-812L6 /7*L2*8\ )8- ,6.36b79J}0M b*18-3 v{l; m} 0 ,-3;67?,M b*18-3 N Φ{A; B} 16/68*7?,-M ` Φ 0;*J8\ /--8b*8/8b*,,-\ 1--3L0,68 M0

M0

{l; m} =

ny

x0 0 ; ∓ b2 a2

o

Ô

n x y o 0 0 {A; B} = ± 2 ; 2 . a b

8/JL6 ,6N-L0; 236b,*,0* 16/68*7?,-M `

M0 Φ

±

y0 x0 (x − x ) + (y − y0 ) = 0 0 a2 b2

⇔

aã

x0 y0 (x − x ) ± (y − y0 ) = 0. 0 a2 b2 (x0 ; y0 )

b 236b,*,09N aã /1-+10 0 2)08b69\ )8- 1--3L0,68 Ç6/13b69 8-)10 M 2L-b7*8b-39J8 236b,*,0J ã_\ .-72)6*;\ )8- 236b,*,0* 16/6K 8*7?,-M 1 ~770./2\[6L6,,-;2 16,-,0)*/10; 236b,*,0*;\b 8-)1* M (x ; y ) 0;**8 b0L 0

0

0

0

x0 x y0 y + 2 = 1, a2 b

6 236b,*,0* 16/68*7?,-M 1 :0.*3+-7*\ [6L6,,-M 16,-,0)*/10; 236b,*,0*;\ b 8-)1* M (x ; y ) 0;**8 b0L 0

0

0

x0 x y0 y − 2 = 1. a2 b D

NM0 D M0

M

O

M0 NM0

O `M0 Φ

`M0 Φ

Ç0/= Ú`=

ðÍµxñÍXW´Z {*18-3 N Φ ± ; ,6[b6J8 861|* º ¦ Φ = ] 236b,*,0* aã 16/68*7?,-M ` Φ ;-|,- .-72)08?\ [6.0/b69 2/7-b0* −→ −−−→ /-.39|*,,-/80 b*18-3-b −OM {x ; y } 0 M M {x − x ; y − y } = Ï³zWñx¶yWx ¶tuÌ¶ztÍ -W³¶Í W VW³xTvuUZ 7*L2J}** /b-M/8b- ~770./-b 0 :0.*3+-7 0/.-7?[2*8/9 b -.801* .30 0[:-8-b7*,00 -836|68*7*M= ðÍwÍñÍ ÊÊZ -16[68?\ )8- 16/68*7?,* ` Φ 1 ~770./6; 0 :0.*3+-76; -+36[2J8 36b,* 2:7 / O-167?,;0 36L02/6;0 F M 0 F M \ .3-b*L*,K ,;0 b 8-)12 16/6,09 = Ô¸ M0

x0 y0 a2 b2

M0

0

0

0

0

0

0

M0

1

0

2

0

M0 `M0 Φ F2

O

F1

P

F2

O

M0 P F1 `M0 Φ

Ç0/= ò^= óyÍêÍXWxZ 79 3*I*,09 ~8-M [6L6)0 /7*L2*8 70+- b)0/708? 1-/0,2/ 2:7-b ;*|L2 ,6.36b79J}0;0 b*18-36;0 16/68*7?,-M 0 O-167?,N 36L0K 2/-b 8*3;0, 4O-167?,M 36L02/< -[,6)6*8 -L,-b3*;*,,- 0 .39;2J F M 0 L70,2 -83*[16 F M \ 70+- ,6M80 1--3L0,68 8-)10 P .*3*/*)*,09 16/6K 8*7?,-M / -/?J 6+/0// 16,-,0)*/1-M /0/8*; 1--3L0,68 0 [68*; b-/.-7?K [-b68?/9 /b-M/8b6;0 +0//*1830/ b,*I,*:- 0 b,283*,,*:- 2:7-b 83*2:-7?K ,016= ðÍwÍñÍ ÊZ -16[68?\ )8- /-O-12/,* 0;*J}0* 8* |* /6;* O-12/ ~770./ 0 :0.*3+-76 .*3*/*16J8/9 .-L .39;; 2:7-;= 0

0

Ç0/= ò= Ô¾

óyÍêÍXWxZ {-/.-7?[-b68?/9 28b*3|L*,0*; .3*LL2}*M [6L6)0= àxyuµxXwSxµÍ´ WzxTÍzSTÍá âä \ :76b6 @@\ åå]\ ÆÅ âòä \ :76b6 _\ åÚ = ðÍwÍñW W S³TÍXxXW´á â]ä \ Úã`\ Úa^\ Úa\ Úa]\ ÚaÆ\ Úaò\ Úaã\ Úa`\ Ú`^\ Ú`\ Ú`]\ Ú`Æ\ ò^Ú\ ò^\ òÚ`\ òò\ òò\ òòò\ òò_\ òòã\ òòa\ òã^\ òã_Å âÚä \ 8*; _\ ã= cc 0plegij mlil2hnj ' H2/8? Φ · .636+-76\ [6L6,,69 16,-,0)*/10; 236b,*,0*; aa y = 2px, 6 ` · .39;69 / ,6.36b79J}0; b*18-3-; u{l; m} \ .3-N-L9}69 )*3*[ 8-)12 ) M (x ; y ) \ [6L6,,69 .636;*830 */10;0 236b,*,09;0 a` x = x + lt, y = y + mt. H3*L.-7-|0;\ )8- .39;69 ` .*3*/*16*8 .636+-72 Φ = 79 ,6N-|L*,09 8-K )*1 .*3*/*)*,09 ` / Φ .-L/86b79*; 236b,*,09 a` b aa= { 3*[27?868* .-72)6*; ^ (y + mt) = 2p(x + lt) ⇔ m t + 2t(my − pl) + y − 2px = 0. ` H30 m 6= 0 236b,*,0* `^ · 1b6L368,-* 236b,*,0* -8,-/08*7?,- t \ .-~8-K ;2 b ~8-; /72)6* .39;69 ` .*3*/*16*8 .636+-72 Φ b Lb2N 8-)16N b-[;-|,/-b.6L6J}0N= H30 m = 0 .39;69 ` .63677*7?,6 -/0 .636+-7= { ~8-; /72K )6* `^ · 236b,*,0* .*3b-M /8*.*,0 0\ /7*L-b68*7?,-\ 0;**8 *L0,/8b*,,-* 3*I*,0* -8,-/08*7?,- t \ .-~8-;2 .39;69 ` .*3*/*16*8 .636+-72 Φ b -L,-M 8-)1* = Ï³TxwxxXWxZ è£¦ ¦ u{l; 0} ¢£ ¡¤© ¡£ ª£ ¡¬ y = 2px ¥ ¡££ ` ¥ ¡º £ ¡¬¥ ¢£ ¡£¦ ¡¤© ¡º £ ª£ ¡¬§ H2/8? ,6.36b7*,0* 36//;6830b6*;-M .39;-M ` ,* 9b79*8/9 6/0;.8-80)*K /10;= -:L6 236b,*,0* `^ 0;**8 Lb6 b*}*/8b*,,N 3*I*,09 ; .3*L.-K 7-|070\ )8- ` 0 Φ .*3*/*16J8/9 t 0 t \ 1-8-3; /--8b*8/8b2J8 Lb* -+}0* 2

0

0

0

0

0

2

0

0

2 2

2 0

0

2

1

2

ÔÁ

0

8-)10 M 0 M 130b-M Φ 0 .39;-M ` = -)16 A \ 9b79J}69/9 /*3*L0,-M N-3K L M M \ /--8b*8/8b2*8 [,6)*,0J .636;*836 t \ 36b,-;2 t = (t + t )/2 = 1

1

2

2

1

A

2

M2 D A u

M1

Ç0/= ò]= H-7-|0; M (x ; y ) = A = -:L6 t = 0 0\ /7*L-b68*7?,-\ t + t = 0 = H8*-3*;* {0*86 b ~8-; /72)6* 1-~OO00*,8 .30 t b 1b6L368,-; 236b,*,00 (90) 36b*, ,27J ` my − pl = 0. 861\ 1--3L0,68 /*3*L0, N-3L M M 2L-b7*8b-39J8 236b,*,0J `= H-/1-7?12 M M · ~8- .3-0[b-7?,69 N-3L6\ 0;*J}69 L6,,-* ,*6/0;.8-K 80)*/1-* ,6.36b7*,0* u{l; m} \ 8- ;,-|*/8b- /*3*L0, b/*N N-3L\ 0;*J}0N ,6.36b7*,0* u = {l; m} \ 2L-b7*8b-39*8 236b,*,0J `] my − pl = 0. (36b,*,0*; `] [6L6*8/9 ,*1-8-369 .39;69\ .63677*7?,69 -/0 .636+-K 7 Φ = Ï³TxwxxXWxZ ££ D(u) ¥ À«£ (92) ¥ ¢£ ªº ¡¬ Φ ¥ ¡£» Ð ª ª© ¡¤© ¡£ u{l; m} ¥ ¥ ¡¥ ª¥ ¡£» ¡º ¤ ¡À u{l; m} § {/* L06;*83 .636+-7 .63677*7?, ** -/0= [ 236b,*,09 `] /7*L2*8 861|*\ )8- b/9169 .39;69 y = y \ .63677*7?,69 -/0 .636+-7\ 9b79*8/9 L06;*83-;\ /-.39|*,,; ,*1-8-3-;2 ,6.36b7*,0J\ 6 0;*,,-\ ,6.36b7*K ,0J u{y , p} = 0

0

0

1

A

A

1

1

2

0

0

ÔÈ

2

2

,Í¶ÍzxëXUx y ³ÍTÍvuxZ

H2/8? 8-)16 M (x ; y ) .30,6L7*|08 .636+-7* Φ = 06;*83 D .636+-7 ) ) Φ \ .3-N-L9}0M *3*[ M (x ; y ) \ 0;**8 236b,*,0* y = y = P61 0 b /72 6* ~770./6 0 :0.*3+-7 /;= /= aÆ\ .39;69 ` Φ \ .3-N-L9}69 )*3*[ M 0 0;*J}69 ,6.36b7*,0*\ /-.39|*,,-* L06;*832 D \ .*3*/*16*8 .636+-72 Φ b Lb2N /-b.6L6J}0N 8-)16N = ðÍµxñÍXWxZ { /72)6* .636+-7 /-N36,9*; 8*3;0,-7-:0J\ 0/.-7?[-b6,K ,2J 36,** L79 ~770./6 0 :0.*3+-7= Ï³TxwxxXWxZ ££ ` Φ ¥ ¡Ðª£«£ ¤¢ ¤ M ¡¬ Φ À«£ ¡¥ ¡£» ª D ¥ ¡Ðª£« ¤¢ Ã ¤¥ ¢£ ¦ Φ ¤ M § H39;69 ` 9b79*8/9 16/68*7?,-M 1 .636+-7* Φ b 8-)1* M 8-:L6 0 8-7?18-:L6\ 1-:L6 8-)16 M 9b79*8/9 Lb-M,-M 8-)1-M .*3*/*)*,09 ` 0 Φ = 16/68*7?,-M ` Φ 1 .636+-7* Φ b 8-)1* M (x ; y ) 6.36b79J}0; b*18-3-; 9b79*8/9 b*18-3 u{y , p} = H-~8-;2 16/68*7?,69 ` Φ 0;**8 /7*L2J}** 236bK ,*,0* `Æ −p(x − x ) + y (y − y ) = 0. b 236b,*,00 `Æ /1-+10 0 2)0K Ç6/13b69 8b69\ )8- 1--3L0,68 (x ; y ) 8-)10 M 2L-b7*8b-39J8 236b,*,0J aa\ .-72)6K M *;\ )8- 236b,*,0* 16/68*7?,-M 1 .636+-K 7*\ [6L6,,-M 16,-,0)*/10; 236b,*,0*;\ b 8-)1* M (x ; y ) 0;**8 b0L ` Φ 0

0

0

0

0

0

0

0

M0

0

M0

0

0

0

0

M0

0

0

0

M0

0

0

0

0

0

0

0

0

0

M0

0

Φ

yy0 = p(x + x0 ).

Ï³zWñx¶yux ¶tuÌ¶ztu ³ÍTÍvuUZ Ç0/= òÆ= 7*L2J}** /b-M/8b- .636+-7 0/.-7?[2*8/9 b -.801* .30 0[:-8-b7*,00 -836|68*7*M= ðÍwÍñÍ ÊZ -16[68?\ )8- 16/68*7?,69 ` Φ 1 .636+-7* Φ b 8-)1* M -+36[2*8 36b,* 2:7 / O-167?,; 36L02/-; F M 0 -/?J .636+-7= óyÍêÍXWxZ H2/8? P · 8-)16 .*3*/*)*,09 16/68*7?,-M / -/?J 6+/0// 16,-,0)*/1-M /0/8*; 1--3L0,68\ 8-:L6 83*2:-7?,01 P F M · 36b,-+*LK 3*,,M= ÔÔ 0

M0

0

0

ðÍwÍñÍ Ê Z -16[68?\ )8- /-O-12/,* 0;*J}0* 8* |* /6;* O-12/ .636+-7\ ,6.36b7*,09 -/*M 1-8-3N .3-80b-.-7-|, 8- */8? .3-80b-K .-7-|, ,6.36b7*,09 -/*M 6+/0// 16,-,0)*/10N /0/8*; 1--3L0,68\ .*K 3*/*16J8/9 .-L .39;; 2:7-;= óyÍêÍXWxZ {-/.-7?[-b68?/9 28b*3|L*,0*; .3*LL2}*M [6L6)0= M0 P `M0 Φ

F Φ

Ç0/= òÚ = àxyuµxXwSxµÍ´ WzxTÍzSTÍá âä \ :76b6 @@\ åÅ âòä \ :76b6 _\ åÚ = ðÍwÍñW W S³TÍXxXW´á â]ä \ _Æ`\ _Ú^\ _Ú\ _Ú`\ _ò^\ _òò\ _ò`\ __^\ __aÅ âÚä \ 8*;6 a= cæ $hnikje ilqkekp 1nnpmol4 pmei2hnj p mlil2hnj ' H-793,69 /0/8*;6 1--3L0,68 ,6 -30*,803-b6,,-M *b170L-b-M .7-/1-/80 E [6L6*8/9 8-)1-M O ∈ E \ ,6[b6*;-M ¡À\ 0 -30*,803-b6,,-M .39;-M -/?J ` ↑ \ .3-N-L9}*M )*3*[ 8-)12 O \ ,6[b6*;2J ¡£¦ À= H-7-K |*,0* 8-)10 M ∈ E -.3*L*79*8/9 Lb2;9 )0/76;0 (ρ; ϕ) \ -.3*L*79*;;0 −→ \ 6 /7*L2J}0; -+36[-; ρ = |−OM | ϕ · 2:-7\ ,6 1-8-3M ,2|,- .-b*3,28? ,6.36b79J}0M b*18-3 a -/0 ` ↑ \ )8-+ *:- ,6.36b7*,0* /-b.67- / ,6.36bK −→ 7*,0*; b*18-36 −OM = .-793,-M /0/8*;-M 1--3L0,68 */8*/8b*,,- 6//-0032*8/9 .36bM -3K 8-,-3;03-b6,,M 3*.*3 {O; i, j} \ :L* i = a/|a| · *L0,0),M ,6.36b79K J}0M b*18-3 .-793,-M -/0= Ç6L02/Kb*18-3 r 8-)10 M /7*L2J}0; -+36K [-; b36|6*8/9 )*3*[ ** .-793,* 1--3L0,68 (ρ; ϕ) r = ρe(ϕ) \ :L* ÔÙ 2

2

2

= H-~8-;2 .39;-2:-7?,* 1--3L0,68 8-)10 M -8K /b9[6, / ** .-793,;0 1--3L0,686;0 /7*L2K ,-/08*7?,- 3*.*36 J}0;0 /--8,-I*,09;0 `Ú x = ρ cos ϕ, y = ρ sin ϕ. [ 236b,*,0M `Ú b0L,-\ )8- .39;-2:-7?,* 1--3L0,68 8-)10 M -L,-K [,6),- -.3*L*79J8/9 .- ** .-793,; 1--3L0,686;= L,61- -+368,-* ,* 0;**8 ;*/86= ¼/70 M 6= O \ 8- ϕ -.3*L*79*8/9 / 8-),-/8?J L- /76:6*;-:-\ 1368,-:- 2πk \ k ∈ Z \ 6 */70 M = O \ 8- ϕ ;-|*8 +8? 7J+; b*}*/8b*,K ,; )0/7-;= P3-;* 8-:-\ 0,-:L6 2L-+,- /)0868?\ )8- ρ ;-|*8 .30,0;68? 0 -83068*7?,* [,6)*,09 = H30 ~8-; /)086*8/9\ )8- 1--3L0,68 (ρ; ϕ) 0 ) (−ρ; ϕ + π) [6L6J8 -L,2 0 82 |* 8- 12 ~8- ;-|,- .-9/,08? /7*L2J}0; -+36[-; )8-+ ,6M80 ,6 .7-/1-/80 8-)12 M / .-793,;0 1--3L0,686;0 (ρ; ϕ) \ ,2|,- .-b*3,28? .-793,2J -/? b-132: .-7J/6 ,6 2:-7 ϕ 0 [68*; -87-|08? ,6 ,*M 8-)12 / 1--3L0,68-M ρ= e(ϕ) = i cos ϕ + j sin ϕ {O; i, j}

M ϕ O

a

`↑

Ç0/= òò= H30 b+-3* .-793,-M /0/8*; 1--3L0,68 L79 0[2)*,09 ~770./6\ :0.*3K +-7 070 .636+-7 */8*/8b*,,- 36/.-76:68? .-7J/ 070 b ,6)67* 16,-,0K )*/1-M /0/8*; 1--3L0,68\ 070 b O-12/* 36//;6830b6*;-M 130b-M\ 6 b 16K )*/8b* .-793,-M -/0 */8*/8b*,,- +368? -/? 6+/0// 16,-,0)*/1-M /0/8*; 1--3L0,68= { ~8-; .636:36O* ; 36//;-830; /7*L2J}2J .-793,2J /0/8*;2 1--3K L0,68\ 2L-+,2J L79 -L,-b3*;*,,-:- 0[2)*,09 0 ~770./6\ 0 :0.*3+-7\ 0 .636+-7 .-7J/ O .-;*}6*; b O-12/ F .636+-7\ 7*bM O-12/ ~770./6 0 b .36bM O-12/ :0.*3+-7 -8,-/08*7?,- 16,-,0)*/1-M /0/8*; 1--3L0K ,68\ 6 b 16)*/8b* .-793,-M -/0 +*3*; -/? 6+/0// 16,-,0)*/1-M /0/8*; 1--3L0,68 /;= 30/= ò_= 610* .-793,* /0/8*; 1--3L0,68 ,6N-L98 .30K ;*,*,0* b 6/83-,-;00\ .-/1-7?12 836*18-309;0 ,*+*/,N 8*7\ .30+70|6K ÙÝ

J}0N/9 1 /-7,*),-M /0/8*;*\ 9b79J8/9 ~770./\ :0.*3+-7 0 .636+-7\ b -L,-; 0[ O-12/-b 1-8-3N ,6N-L08/9 /-7,* ,*+*/,* 8*76 36//;6830b6K J8/9 161 ;68*3067?,* 8-)10= H2/8? D · L03*1830/6\ /--8b*8/8K M b2J}69 O-12/2\ 36/.-7-|*,,-;2 b .-K L 7J/* = ¹-167?,; .636;*83-; 130b-M ϕ ,6[b6*8/9 .-7-b0,6 O-167?,-M N-3 K Φ L\.*3.*,L012793,-M -/0 130b-M= +-K K F =O ρ [,6)6*8/9 ~8-8 .636;*83 +21b-M p = )*K b0L,-\ O-167?,M .636;*83 /-b.6L6*8 D / O-167?,; 36L02/-; 8-)10 A ∈ Φ ò_= 0/ Ç = − → L79 1-8-3-M b*18-3 F A .*3.*,L01279K 3*, -/0 6+/0// 16,-,0)*/1-M /0/8*; 1--3L0,68= +-[,6)0; /0;b-7-; δ = dist(F, D) 36//8-9,0* -8 O-12/6 L- L03*1830/\ 8-:L6 p = δe \ :L* e · ~1/*,830/08*8= { /72)6*) .636+-7 O-167?,M .636;*83 /-b.6L6*8 / .636;*83-; p b ** 16,-,0 */1-; 236b,*,00\ 6 L79 ~770./6 0 :0.*3+-7 p = b /a = H2/8?\ L67**\ K · 8-)16 .*3*/*)*,09 .-793,-M -/0 0 L03*1830/ D \ M ) Φ \ L · .3-*109 8- 10 M ,6 L03*1830/2\ · .3-0[b-7?,69 8-)16 130b-M ) r · O-167?,M 36L02/ 8- 10 M \ d · 36//8-9,0* -8 M L- L03*1830/ = ;**; 2

`ò = 0 + δ + (re(ϕ), i) = δ + r cos ϕ. H-L/86b799 b36|*,0* `ò L79 d b 236b,*,0* r = de \ -L,-[,6),- / 8-),-K /8?J L- 36/.-7-|*,09 ,6 .7-/1-/80 -.3*L*79J}** 130b2J Φ \ .-72)6*; ) r = δe + re cos ϕ \ -812L6 r(1 − e cos ϕ) = δe = p = 8/JL6 .-72 6*; 236bK ,*,0*\ [6L6J}** .30 /--8b*8/8b2J}0N [,6)*,09N ~1/*,830/08*86 ~770./\ .636+-72 070 .36b2J b*8b? :0.*3+-7 p `_ . ρ= 1 − e cos ϕ −−→ −−→ −−→ −−→ −−→ d = |LM | = (LM , i) = (LK + KF + F M , i) =

ÙÜ

ðÍwÍñÍ ÊZ {b*/80 /7*L2J}** 236b,*,0* 7*b-M b*8b0 :0.*3+-7 b ~8-M |* .-793,-M /0/8*;* 1--3L0K ,68 .-7J/ ,6N-L08/9 b .36b-; O-K 12/*\6 .-793,69 -/? /-b.6L6*8 / -/?J 6+/0// 16,-,0)*/1-M /0/8*; 1--3K L0,68

M

L ρ

K F1 = O

Ç0/= òã=

−p . ρ= 1 + e cos ϕ

D

àxyuµxXwSxµÍ´ WzxTÍzSTÍá âä \ :76b6 @@\ åòÅ âòä \ :76b6 _\ åÆ= ðÍwÍñW W S³TÍXxXW´á â]ä \ ^_\ ^ã\ ^`\ \ ]\ Ú\ Æã^\ Æã\ Æã]\ ÆãÆ\ ÆãÚ\ Æãò\ Æã_\ Æa^\ Æa]\ ÆaÆ\ ÆaÚ\ Æaò\ ÆaaÅ âÚä \ 8*;6 = c $lilegip+eofpe ilqkekp 1nnpmol4 pmei2hnj p mlil2h nj ' H636;*830)*/10; 236b,*,0*; 70,00 Γ ⊂ A b 6OO0,,-; .3-/836,/8b* ,6K [b6J8 236b,*,0* b0L6 r = r(t) \ :L* t · b*}*/8b*,,M .636;*83\ .30,0;6K J}0M [,6)*,09 0[ ,*1-8-3-:- 0,8*3b676 (α; β) ⊂ R \ 6 r(t) · 36L02/Kb*18-3 ,*1-8-3-M 8-)10 70,00 Γ .30 7J+-; [,6)*,00 t ∈ (α; β) = +76/8? [,6)*,0M O2,100 r(t) ;-|*8 ,* /-b.6L68? /- b/*M 70,0*M Γ \ 8- */8? ~86 O2,109 ;-|*8 .636;*830[-b68? 8-7?1- )6/8? 70,00= { [6b0/0;-/80 -8 3*I6*;-M [6L6)0 .3*L.-76:6*8/9 861|*\ )8- ~86 O2,109 2L-b7*8b-39*8 ,*1-8-3; L-.-7,08*7?,; 2/7-b09;\ ,6.30;*3 9b79*8/9 0,*180b,-M 070 +0*180bK ,-M\ ,*.3*3b,-M 070 12/-),-KL0OO*3*,032*;-M 0 8= .= )*b0L,-\ .636K ;*830)*/1-* 236b,*,0* 70,00 -.3*L*79*8/9 ,*-L,-[,6),- = 6.30;*3\ */70 ) r = r(t) · .636;*830 */1-* 236b,*,0* ,*1-8-3-M 70,00 Γ ⊂ A \ 6 t = t(τ ) · ,*1-8-369 b[60;,- -L,-[,6),69 O2,109\ (γ; δ) 3 τ )7→ t(τ ) ∈ (α; β) \ 8236b,*,0* r = r(τ ) = r(t(τ )) 861|* 9b79*8/9 .636;*830 */10; 236b,*,0*; 70,00 Γ = ¼/70 b .3-/836,/8b* A b+36, 3*.*3 {O; e } \ 8-\ 36[7-|*,09 r(t) = ) x (t)e -.3*L*79J8 1--3L0,68,* .636;*830 */10* 236b,*,09 x = x (t) \ Ùï n

n

n

i

i

i

i

i

\ 70,00 Γ = ,0 861|* -.3*L*7*, ,*-L,-[,6),-\ [6b0/98 -8 b+-36 236b,*,09 r = r(t) 0 -8 b+-36 3*.*36= :0.*3+-72 0 .636+-72 ,6 .7-/1-/80 E 861|* ;-|,- [6L6b68? 770./\ 36[70),;0 .636;*830)*/10;0 236b,*,09;0= 6.30;*3\ ;-|,- .-72)08? .636;*830)*/10* 236b,*,09 ~80N 70,0M\ .*3*N-L9 b 0N .-793,N 236b,*,0K 9N `_ -8 .-793,N 1--3L0,68 1 .39;-2:-7?,; .- O-3;276; x = ρ cos ϕ \ { ) ) ) y = ρ sin ϕ = 16 */8b* .636;*836\ [6L6J}*:- 8- 12 .636+-7\ - *b0L,-\ ;-|,- b[98? ** -3L0,682 x = t /2p \ y = t = { ~8-; .636:36O* 36//;6830b6J8/9 .636;*830)*/10* 236b,*,09 ~770.K /6 0 :0.*3+-7\ b 1-8-3N .636;*83\ [6L6J}0M 8-)12\ .30,6L7*|6}2J 70,00\ 0;**8 */8*/8b*,,M :*-;*830)*/10M /;/7 = /,6)676 -132|,-/8? x +y = Ç6//;-830; y ) a = - 16 M ~8-M -132|,-/80 0;**8 36L02/K b*18-3 r = ae(t) \ :L* t · 2:-7\ ,6 1-8-3M M ,6L- .-b*3,28? ,6.36b79J}0M b*18-3 i = j ) t e -/0 6+/0//\ 8-+ *:- ,6.36b7*,0* /-bK .67- / ,6.36b7*,0*; 36L02/6Kb*18-36 8-)10 O A x i M \6 e(t) = i cos t+j sin t · 132:-b69 b*18-3K ,69 O2,109 /;= 30/= òa= 610; -+36[-;\ -132|,-/8? ;-|*8 +8? [6L6,6 .636;*830K )*/10;0 236b,*,09;0 Ç0/= òa= `ã r = ae(t) ⇐⇒ x = a cos t, y = a sin t. +76/8? )(α; β) 0[;*,*,09 .636;*836 t b+036*8/9 b [6b0/0;-/80 -8 2/7-K b0M [6L6 0= 6.30;*3\ */70 8-)16 M (t) ,*.3*3b,- b36}6*8/9 b-132: 8-)K 10 O \ ;-|,- /)0868?\ )8- t ∈ (−∞; +∞) = H636;*83 t b 236b,*,09N `ã 0;**8 861|* /7*L2J}0M :*-;*830)*/10M /;/7 )0/7- a t · ~8- 2Lb-*,,69 -30*,803-b6,,69 .7-}6L? /*18-36 OAM \ :L* A(a; 0) · 8-)16 .*3*/*)*,09 -132|,-/80 / .-7-|08*7?,-M )6/8?J -/0 6+/0//= { -/,-b* .636;*830)*/10N 236b,*,0M -132|,-/80 `ã 7*|08 830:-,-K ;*830)*/1-* 8-|L*/8b- cos t + sin t = 1 = 8- |* 8-|L*/8b- .-[b-79*8 i = 1, . . . , n

2

2

2

2

1

2

2

2

Ù

2

.636;*830[-b68? ~770./

`a

x2 y 2 + =1 a2 b2

/7*L2J}0; -+36[-;

x = a cos t,

``

y = b sin t.

Q*-;*830)*/10M /;/7 .636;*836 t b y 236b,*,09N `` 9/*, 0[ 30/2,16 ò`\L*K ;-,/83032J}*:- 8-8 O618\ )8- ~770./ b `a ;-|*8 +8? .-72)*, 0[ -132|,-K M y 5 /80 36L02/6 a /|680*; 1 -/0 6+/0// / 1-~OO00*,8-; b/a 070 36/89|*,0*; x a x -132|,-/80 36L02/6 b -8 -/0 -3L0,68 / 1-~OO00*,8-; a/b = {)0/70; 8*.*3? L79 ~770./6 .7-K }6L? S(t) /*18-36 OAM \ :L* A(a; 0) · 0/= ò`= Ç 8-)16 .*3*/*)*,09 ~770./6 / .-7-|0K 8*7?,-M )6/8?J -/0 6+/0// b*3I0,6 ~770./6\ 6 M (t) · 8-)16 ~770.K /6\ -.3*L*79*;69 /--8b*8/8b2J}0; [,6)*,0*; .636;*836 t = H3-0[b-L,69 O2,109 S (t) .3*L/86b79*8 .3*L*7 -8,-I*,09 .7-}6L0 /*18-36 OM M \ :L* M · 8-)16 ~770./6\ /--8b*8/8b2J}69 [,6)*,0J .636;*836 t + ∆t \ 1 ) H ∆t .30 ∆t → 0 = -/1-7?12 .7-}6L? /*18-36 OM M -870 6*8/9 -8 .7-K }6L0 83*2:-7?,016 OM M ,6 +*/1-,*),- ;672J b*70)0,2 +-7** b/-1-:.-39L16 )*; ∆t \ 8- .3-0[b-L,69 S (t) 36b,6 .3*L*72 -8,-I*,09 .7-}6L0 83*2:-7?,016 OM M 1 ∆t .30 ∆t → 0 = M

M

0

0

0

0

0

0

0

M0

M (t) M O

O

Ç0/= _^=

Ç0/= _= Ù¸

A

861\ S 0 (t) = lim

8/JL6

∆t→0

a cos t b sin t 1 2 a cos(t + ∆t) b sin(t + ∆t) ∆t

=

ab sin ∆t ab lim = . 2 ∆t→0 ∆t 2

^^

^ 610; -+36[-;\ .636;*83 t b 236b,*,09N `` · ~8- -30*,803-b6,,69 .7-K }6L? /*18-36 OAM \ b[9869 / 1-~OO00*,8-; 2 = Q0.*3+-70)*/10* O2,100 ch t = (e + e ab) 0 sh t = (e − e ) 2L-K b7*8b-39J8 8-|L*/8b2 ch t − sh t = 1 = 8- 8-|L*/8b- L6*8 b-[;-|,-/8? .636;*830[-b68? .36b2J b*8b? :0.*3+-7 y x ^] − =1 a b /7*L2J}0; -+36[-; ^Æ x = a ch t, y = b sh t. Q*-;*830)*/10M /;/7 .636;*836 t b 236b,*,09N ^Æ 6,67-:0)*, :*-;*8K 30)*/1-;2 /;/72 .636;*836 t b 236bK ,*,09N ~770./6 ``= {)0/799 .3-K M 0[b-L,2J S (t) -8 O2,100 S(t) \.3*LK A /86b79J}*M /-+-M .7-}6L? /*18-36 O OAM \:L* A(a; 0) · b*3I0,6 :0.*3+-K 7\ 6 M (t) · 8-)16 :0.*3+-7\ -.3*K L*79*;69 /--8b*8/8b2J}0; [,6)*,0K *; .636;*836 t \ 6,67-:0),- /72)6J Ç0/= _^= ~770./6 .-72)0; S(t) =

Z

t

0

S (t) dt =

0

2

Z

t

0

2

1 2

2

2

2

2

ab ab dt = t. 2 2

−t

t

1 2

t

−t

0

S 0 (t) = lim

∆t→0

a ch t b sh t 1 2 a ch (t + ∆t) b sh (t + ∆t) ∆t

=

ab sh ∆t ab lim = , 2 ∆t→0 ∆t 2

^Ú

-812L6 6,67-:0),- ^ .-72)6*; S(t) = ab t = 610; -+36[-;\ .636;*83 t 2 Ù¾

b 236b,*,09N ^Æ · ~8- -30*,803-b6,,69 .7-}6L? /*18-36 OAM \ -:36K ,0)*,,-:- :0.*3+-7-M\ b[9869 / 1-~OO00*,8-; 2 = ðÍwÍñÍ ÊZ /.-7?[-b6,0* 830:-,-;*830)*/1-:-ab8-|L*/8b6 sec t−tg t = ) ) 1 .-[b-79*8 .-72 08? /7*L2J}0* .636;*830 */10* 236b,*,09 L79 .36b-M b*8b0 :0.*3+-7 ^] π π ^ò x = sec t, y = tg t, t∈ − , . 2 2 ) /.-7?[29 Lb* 1-,*,830 */10* -132|,-/80 36L02/-b a 0 b \ 161 ,6 30/2,1* ò`\216[68? 8-)12 :0.*3+-7\ /--8b*8/8b2J}2J L6,,-;2 [,6)*,0J .636;*836 t = ] ¯6.0/68? 6,67-:0),* .636;*830)*/10* 236b,*,09 L79 7*b-M b*8b0 :0.*3+-7 ^]= ðÍwÍñW Wê -yêÍµxXÍÎWuXXUõ vWxzut XÍ ¶tuÌ¶ztÍ -W³¶Í& VW³xTvuU W ³ÍTÍvuUá âaä \ Ú^\ Ú_\ Úã\ ÚÚ^\ ÚÚ^6\ ÚÚ`\ Ú_ã\ Ú_a\ Ú_`\ Úã^\ ò^\ ò^]\ ò^Æ\ ò^ã= 2

2

mpohf npgeilgij âä 57*1/6,L3-b H== ¤¦ © ¦¦ ©¬= °= 6216= `ã`= ò] /= â]ä Â6Nb67-b ={=\ °-L*,-b H==\ H63N-;*,1- 5 == ú¬ ¢ª¤ ¡ º ¤¦ ©= °= 6216= `_Ú = âÆä ¼O0;-b ={=\ Ç-[*,L-3, =Ç= ô¦£ ©¬ ©£ ©º £= °= 6216= `ã^= ò]a /= âÚä :2L*/;6, P=Â= ª¤ ¡ ¤¦ ©§ 7 8= ()*+K ,-* .-/-+0* 1 123/2 45,67080)*/169 :*-;*8309<= P6[6,/1= 2,K8= ]^^Æ= _Æ /= ÙÁ

âòä 7?0, {=5 =\ H-[,91 =Q= ¤£ ©£= °= 6216= `a= ]Æ] /= â_ä P23-I 5 =Q= Ä¦ ©¬= °= 6216= `ã= ÚÆ] /= âãä H-/8,01-b °=°= ¤£ ©£ ( ôÑ ¡ ©§ ú 8 ) = °= 6216= `ã`= ÆÆ_ /= âaä 92+*3+077*3 == ª¤ ¡»£ ¡ ¤¦ ©= °= 6216= `_Ú = ÆÆ_ /=

ÙÈ

hçei:lkpe {*18-3 ,6 .7-/1-/80 0 b .3-/836,/8b* ========================= Æ P--3L0,68 b*18-3-b =========================================== 5OO0,,* /0/8*; 1--3L0,68 ,6 .7-/1-/80 0 b .3-/836,/8b* ====================================== ]Æ 16793,-* .3-0[b*L*,0* b*18-3-b ============================== ]` .*3609 .-b-3-86 b*18-36 ,6 2:-7 α ========================== ÚÆ P-/-* .3-0[b*L*,0* b*18-3-b ,6 .7-/1-/80 ===================== Úã H39;69 70,09 ,6 6OO0,,-M .7-/1-/80 ========================= Ú` H39;69 70,09 ,6 *b170L-b-M .7-/1-/80 ======================== ò_ P-,0)*/10* /*)*,09 ============================================= _ 06;*83 ~770./6 0 :0.*3+-7 ================================ ã` 06;*83 .636+-7 ============================================ a_ H-793,* 236b,*,09 ~770./6\ :0.*3+-7 0 .636+-7 ========== a` H636;*830)*/10* 236b,*,09 ~770./6\ :0.*3+-7 0 .636+-7 ================================ `]

ÙÔ

I

I Miserabili - I

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Algebra I: Pt. I

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А.Насибов. I-W-I

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I Did Tell, I Did

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I Rant, Therefore I Am

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I Write What I Like

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I vecchi e i giovani

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