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ПРЕДАВАЊА ПО ВЕКТОРСКА И ЛИНЕАРНА АЛГЕБРА ПОДГОТВИЛ Ристо Малчески Скопје, 2007

Risto Malceski Predavanja Po Vektorska i Linearna Algebra1

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Text of Risto Malceski Predavanja Po Vektorska i Linearna Algebra1

, 2007

,

I 1. . 2. 3. 4. 5. 6. II 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. . 29 31 34 36 37 39 40 41 44 47 49 56 59 62 68 69 74 76 79 80 7 10 12 15 18 25

III 1. 2. 3. 4. 5. 87 87 89 91 92

3

6. 7. . 8. . 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. IV 1. . 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. -

93 97 99 100 101 105 111 113 118 123 126 129 130 132 134 135

141 145 152 156 159 162 168 170 177 181 189 194 199 202 204 207 210 212 213 214 219

4

22. 23. . 24. 25. 26. . 27. 28. 29. . 30. 31. 32. 33. . 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. . 46. 47. 48. 1. . 2. . 3.

224 228 231 234 237 243 248 253 255 258 262 270 273 276 281 285 287 289 292 301 305 310 312 314 321 325 329

351 353 358 361

5

6

I 1. . 1.1. . a1 , a2 , b1 , b2 :

1 a2

b1 . b2

(1)

a1b2 a2b1 (1). 1 a2 b1 = a1b2 a2b1. b2

(2)

a1 , a2 , b1 , b2 . a1 , b2 , a2 , b1 . , .1.2. .

5 3 = 5i4 3i(1) = 23. 1 41.3.

a1 x + b1 y = d1 a2 x + b2 y = d 2 ,

(3)

x, y . x0 , y0 (3), x0 , y0 , .. x, y x0 , y0 (3) .

(3) b2 , b1 (a1b2 a2b1 ) x = d1b2 d 2b1 . (4)

7

(a1b2 a2b1 ) y = a1d 2 a2 d1 . = 1 a2 b1 d , x = 1 b2 d2 b1 , y = 1 b2 a2 d1 . d2

(5)

(6)

, (4) (5) i x = x , i y = y . (7)

. , (3) (3).

x (3), y (3). , 0 . (7) x = x , y = y .

(8)

(8) (3), .. .. (3) , , (8). .

3x + 4 y = 2 2 x + 3 y = 7..

=

3 4 2 3

= 9 8 = 1.

0 , , (8). 2 4 3 2 x = = 6 28 = 22, y = = 21 4 = 17. 7 3 2 7 ,22 x = x = 1 = 22, y = y = 17 = 17. 1

8

1.4. = 0 , x , y . , (7) , (7) . , (3) . , (7) (3), (3), , (7). , . . = 0 , x , y , (3) , .. . .

3x + 4 y = 1 6 x + 8 y = 3..

= x = 1 4 3 8

3 4 6 8

= 24 24 = 0 ,

= 8 12 = 4 0, y =

3 1 6 3

= 9 6 = 3 0,

. 1.5. = x = y = 0 .

a1b2 a2b1 = 0, a1d 2 a2 d1 = 0, d1b2 d 2b1 = 0, .. a1 : a2 = b1 : b2 = d1 : d 2 , (3) . , (3) , .. a1x + b1 y = d1. (9) , (9) , x, y . , .. = x = y = 0 , (3)

. 9

. , (3), , . , , . ,

0i x + 0i y = 4 0i x + 0i y = 4 = x = y = 0 , , .. ..

3x + 4 y = 1 6 x + 8 y = 2.. = x = y = 0 2, . , y 3 x y = 14 x , 3 A = {( x, 14 x ) | x R} .

1.6. a1 x + b1 y = 0 (10) a2 x + b2 y = 0,.. , . , (10) x = y = 0 . 0 , , = 0 , (10), , , .. .

. (10) = 0 .

2. 2.1. a1x + b1 y + c1 z = 0 a2 x + b2 y + c2 z = 0, x, y, z . 10

(1)

a1 a2

b1 0. b2

(2)

(1) a1x + b1 y = c1 z a2 x + b2 y = c2 z , (3)

, z . , z (3) ,

x=

c1z b1 c2 z b2 a1 b1 a2 b2

, y=

a1 c1z a2 c2 z a1 b1 a2 b2

.

(4)

, x, y z (1), z (1). (4) . , c1 z b1 b1 = c2 z b2 b2 b 1 = 1 b2 c1 a , 2 = 1 c2 a2z

c1 a z, 1 c2 a2

a c1 z = 1 c2 z a2 c1 a , 3 = 1 c2 a2 z

c1 z, c2 b1 b2

(4) x = 1 , y = 2 . 3 3 , z 3

(5)z = 3t

= t , ..

x = 1t , y = 2t . , .

. x = 1t , y = 2t , z = 3t , t R

(6)

(1).

2.2. . 3x + 5 y + 8 z = 0 7 x + 2 y + 4 z = 0.

11

. 1 = 4, 2 = 44, 3 = 29. , A = {(4t , 44t , 29t ) | t R} . 2.3. 3 0 . 3 = 0 , 1 , 2 , . 1 = 2 = 3 = 0 , ..

b1c2 b2c1 = 0, a1c2 a2 c1 = 0, a1b2 b1a2 = 0., (1) , . , 1 = 2 = 3 = 0 , . , , , .

2.4. . 5 x + y 2 z = 0 15 x + 3 y 6 z = 0.

. 1 = 2 = 3 = 0 ( 3). y = 2 z 5 x , x, z R ,..

A = {( x, 2 z 5 x, z ) | x, z R} .

3. 3.1. . a1 , a2 , a3 , b1 , b2 , b3 , c1 , c2 , c3 : a1 a2 a 3

b1 b2 b3

c1 c2 . c3

(1)

12

b a1 2 b3

c2 a b1 2 c3 a3

c2 a + c1 2 c3 a3

b2 b3

(1).

a1 = a2 a3

b1 b2 b3

c1 b c2 = a1 2 b3 c3

c2 a b1 2 c3 a3

c2 a + c1 2 c3 a3

b2 . b3

(2)

a1 , a2 , a3 , b1 , b2 , b3 , c1 , c2 , c3 (1).

3.2. . (2) , a1 a2 a3 b1 b2 b3 c1 c2 = a1b2 c3 + b1c2 a3 + c1a2b3 c1b2 a3 a1c2b3 b1a2 c3 . c3(3)

(3) , : ) , ) +, -.

3.3. . 1 2 3 2 3 4. 4 5 6 (4)

. , :1 2 3 1 2 2 3 4 2 3 = 1i3i6 + 2i4i4 + 3i2i5 3i3i 4 1i 4i5 2i 2i6 4 5 6 4 5 = 18 + 32 + 30 36 20 24 = 80 80 = 0.

3.4. . . -

13

.

3.5. . (4) 5, = 1 3 2 4 = 4 6 = 2 .

3.6. . . , , . , +, , . 3.7. . 3.4 3.6 : M , i j , A = (1)i + j M . , b2 (2) : a B2 = (1)2+ 2 1 a3 c1 . c3

3.8. ( ). Ai , i = 1, 2,3 ai , i = 1, 2,3 , Bi , i = 1, 2,3 bi , i = 1, 2,3 Ci , i = 1, 2,3 ci , i = 1, 2,3 . , = a1 A1 + a2 A2 + a3 A3 , = b1B1 + b2 B2 + b3 B3 , = c1C1 + c2C2 + c3C3 , = a1 A1 + b1B1 + c1C1 , = a2 A2 + b2 B2 + c2C2 , = a3 A3 + b3 B3 + c3C3 . (5)

. 3.1 3.7. . .

14

4.

4.1. . , , , ..a1 a2 a3 b1 b2 b3 c1 a1 a2 b2 c2 a3 b3 . c3 (1) c2 = b1 c3 c1

. (1) , (1) , a1 a2 a3 a1 a2 b1 c1 b2 c2 a3 b b3 = a1 2 c2 c3 b3 b b b b a2 1 3 + a3 1 2 . c3 c1 c3 c1 c2 (3) b1 b2 b3 b c2 = a1 2 b3 c3 c1 c2 b a2 1 c3 b3 c1 b + a3 1 c3 b2 c1 c2 (2)

, (2) (3) (1).

4.2. . , , . . , , a1 = a2 a3 a2 * = a1 a3 a2 * = a1 a3 b2 b1 b3 c2 b1 b2 b3 b2 b1 b3 c1 c2 c3 c2 c1 . c3 (4)

,

* b c1 = a1 ( 2 b3 c3 c2 ) + b1 c3 a2 a3 c2 a + c1 ( 2 c3 a3 b2 ) b3

15

b = (a1 2 b3

c2

a b1 2 c3 a3

c2

a + c1 2 c3 a3

b2 b3

) = ,

.

4.3. . () , . . , ..a1 b1 c1 c1 . c3 = a1 b1 a3 b3

, = * , * . 4.2 * = * = = , .. = 0 .

4.4. . k () , = k 1 , 1 () k . . k . , ka1 kb1 kc1 a2 a3 b2 b3 a1 b1 b2 b3 c1 c2 . c3 c2 = ka1 A1 + kb1B1 + kc1C1 = k (a1 A1 + b1B1 + c1C1 ) = k a2 c3 a3

4.5. . () , . . . , 4.4 4.2 ka1 kb1 kc1 a1 a3 b1 b3 c1 = k c3 a1 b1 c1 c1 = k i0 = 0 . c3 a1 b1 a3 b3

4.6. . () , , () , , .16

. , , .

a1 + 1 b1 + 1 c1 + 1 a2 b2 c2 = (a1 + 1 ) A1 + (b1 + 1 ) B1 + (c1 + 1 )C1 a3 b3 c3 = (a1 A1 + b1B1 + c1C1 ) + (1 A1 + 1B1 + 1C1 )

a1 = a2 a3

b1 b2 b3

c1

1 1 1b2 b3 c2 . c3

c2 + a2 c3 a3

4.7. . , () (). . 4.6 4.5. 4.8. .

1 2 3 = 2 1 2. 3 2 1. 2 , 3 ,

1 2 3 1 0 0 3 4 = 2 1 2 = 2 3 4 = 1i = 24 16 = 8. 4 8 3 2 1 3 4 84.9. (4) , , :

= a1 A1 + a2 A2 + a3 A3 .

(5)

(5) a1 , a2 , a3 d1 , d 2 , d3 . , (5) d1 , d 2 , d3 :

d1 d1 A1 + d 2 A2 + d3 A3 = d 2 d3

b1 b2 b3

c1 c2 . c3

(6)

17

d1 = b1 , d 2 = b2 , d3 = b3 d1 = c1 , d 2 = c2 , d3 = c3 . , (6) , , .. : b1 A1 + b2 A2 + b3 A3 = 0 , c1 A1 + c2 A2 + c3 A3 = 0 . (7) (8)

, a1B1 + a2 B2 + a3 B3 = 0 , c1B1 + c2 B2 + c3 B3 = 0 , a1C1 + a2C2 + a3C3 = 0 , b1C1 + b2C2 + b3C3 = 0 . , . (9) (10) (11) (12)

. () () .

5. 5.1.

a1x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d 2 a x + b y + c z = d , 3 3 3 3

(1)

x, y, z , a1 , a2 , a3 , b1 , b2 , b3 , c1 , c2 , c3 d1 , d 2 , d3 . x0 , y0 , z0 (1), (1), .. x, y, z x0 , y0 , z0 , , .5.2. (1). , a1 b1 b2 b3 c1 c2 c3

= a2 a318

(2)

, (1). A1 , A2 , A3 , B1 , B2 , B3 , C1 , C2 , C3 a1 , a2 , a3 , b1 , b2 , b3 , c1 , c2 , c3 . A1 , A2 , A3 (a1A + a2 A2 + a3 A3)x + (b1A + b2 A2 + b3 A3) y + (c1A + c2 A2 + c3 A3)z = d1A + d2 A2 + d3 A3 . 1 1 1 1 4.9 i x = d1 A1 + d 2 A2 + d3 A3 . i y = d1B1 + d 2 B2 + d3 B3 , i z = d1C1 + d 2C2 + d3C3 . (4) (5) (3)

(3), (4) (5) x , y , z , . , (3), (4) (5) : i x = x , i y = y , i z = z , d1 b1 b2 b3 c1 a1 d1 d2 d3 c1 a1 b1 b2 b3 d1 d2 . d3

(6)

x = d2 d3

c2 , y = a2 c3 a3

c2 , z = a2 c3 a3

(7)

, 0 . , (6) :x = x , y = y , z = z .1 1 a1 x + b1 y + c1 z = a1 (d1 A1 + d 2 A2 + d3 A3 ) + b1 (d1B1 + d 2 B2 + d3 B3 ) 1 + c1 ( d1C1 + d 2C2 + d3C3 ) 1 1 = d1 (a1 A1 + b1B1 + c1C1 ) + d 2 (a1 A2 + b1B2 + c1C2 ) 1 + d3 (a1 A3 + b1B3 + c1C3 )

(8)

(8) (1).

(9)

= a1 A1 + a2 A2 + a3 A3 , 4.9 a1 A2 + b1B2 + c1C2 = a1 A3 + b1B3 + c1C3 = 0

19

(9) a1 x + b1 y + c1 z = d1 , .. x, y, z (8) (1). (1). .. 0 , (1) , (8). 5.3. . , .. (8), . 5.4. .

x + 2 y + 3z = 1 2 x + 3 y + z = 0 3x + y + 2 z = 0. .

1 2 3 = 2 3 1 = 18 0 , 3 1 2 , (8). , 1 2 3 1 1 3 1 2 1 x = 0 3 1 = 5, y = 2 0 1 = 1, z = 2 3 0 = 7 . 0 1 2 3 0 2 3 1 0 , (8) 5 7 1 x = x = 18 , y = y = 18 , z = z = 18 .

5.5. . = 0 , x , y , z , (1) , .. . . , = 0 , x , y , z , (6)

, .. (6) . , (1) -

20

. , (6) (1), (1) (6). 5.6. .

x + y + z = 2 3x + 2 y + 2 z = 1 4 x + 3 y + 3z = 4 = 0 y = 1 0 , 5.5 . 5.7. .

a1x + b1 y + c1 z = 0 a2 x + b2 y + c2 z = 0 a x + b y + c z = 0. 3 3 3

(10)

, x = y = z = 0 (10) .5.8. . 0 , (10) . . 5.2 4.5. 5.9. , = 0 (10) , ,

a1 a2

b1 0. b2

(10) , 2.1 b x= 1 b2 c1 a t, y = 1 c2 a2 c1 t, z = c2 a1 a2 b1 t, t R . b2 (11)

, x, y, z b a3 x + b3 y + c3 z = (a3 1 b2 c1 a b3 1 c2 a2 c1 a + c3 1 c2 a2 b1 )t = it = 0 . b2

, (11) (10). , . , (10) 21

, (10) . , (10) , . , . . (10) =0. 5.10. . x + 2 y + z = 0 3x 5 y + 3 z = 0 2 x + 7 y z = 0, = 33 0 , x= y = z =0. 5.11. . x + y + z = 0 2 x + 3 y + 2 z = 0 4 x + 5 y + 4 z = 0, = 0 . , 1 1 = 1 0 , 5.9 2 3 (11). : x = t , y = 0, z = t , t R . 5.12. .

x + y + z = 0 2 x + 2 y + 2 z = 0 3x + 3 y + 3 z = 0, , = 0 . x + y + z = 0 . {( x, y , z ) | z = x y, x, y R} . 5.13.

22

a1x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d 2 a x + b y + c z = d , 3 3 3 3

(12)

x0 , y0 , z0 (12). x, y, z x0 , y0 , z0 , a1 x0 + b1 y0 + c1z0 = d1 a2 x0 + b2 y0 + c2 z0 = d 2 a x + b y + c z = d . 3 3 0 3 0 3 0 (13)

(12) (13) a1 ( x x0 ) + b1 ( y y0 ) + c1 ( z z0 ) = 0 a2 ( x x0 ) + b2 ( y y0 ) + c2 ( z z0 ) = 0 a3 ( x x0 ) + b3 ( y y0 ) + c3 ( z z0 ) = 0. x x0 = u , y y0 = v, z z0 = w (14) a1u + b1v + c1w = 0 a2u + b2 v + c2 w = 0 a3u + b3v + c3 w = 0. (16) (15) (14)

(16) u , v, w (12). (16) (12). = 0 . , 5.9 (16) , .. . , (12) , (15) : x = x0 + u, y = y0 + v, z = z0 + w u , v, w (16). .. = 0 x = y = z = 0 , (12)

. 5.14. . 23

x + y + z = 1 2 x + 2 y + 2 z = 3 3x + 3 y + 3z = 4 = x = y = z = 0 , . , 2 0i x + 0i y + 0i z = 1 . 5.15. .

3x + y z = 1 5 x + 2 y + 3z = 2 8 x + 3 y + 2 z = 3 = x = y = z = 0 . , , , 3x + y z = 1 5 x + 2 y + 3z = 2 (*)

(*) : 3x + y = 1 + z 5 x + 2 y = 2 3z z , .. . x=1+ z 1 23 z 2 3 1 5 2

= 5z, y =

3 1+ z 5 2 3 z 3 1 5 2

= 1 14 z .

, : {( x, y , z ) | x = 5 z , y = 1 14 z , z R} .

24

6. 1. )3 1 x +1 y +11t 2 1+t 2 2t 1+t 2

2 , 5 y 1 , x 12t 1+ t 2 1t 1+ t 22

)

3 9 a b

2 , 6 b , a logb a , 1 sin . cos

)

a c

a2 c2

,

)

)

)

a2 ab

ab b2

,

)

,

)

1 log a b cos sin

) 2.

sin sin

cos , cos

)

, x : )x +1 3 2 = 0, 4

)

x +1 2

x3 =0. 2x 5 x 7 y = 1 ) x 2 y = 0ax + by = c ) 2 2 bx ay = c, a + b 0

3.

:2 x + 5 y = 1 ) 3x + 7 y = 2 4 x + 7 y + 13 = 0 ) 5 x + 8 y + 14 = 03ax 4by = 2c ) 2ax 3by = c, ab 0

2 x 3 y = 4 ) 4 x 5 y = 10 3 x 7 y = 3 ) x + 4 y = 5

4.

, 4 x + 6 y = 2 ) 6 x + 9 y = 3 ax + 4 y = 2 ) 9 x ay = 3 3x 2 y = 2 ) 6 x 4 y = 3 ax 9 y = 6 ) 10 x by = 10. ax + by = ad ) bx + cy = bd

5.

2 1 3 4 3 2, 3 3 2 5 4 1 3, 2 3 2 1 2 4 2 3

) 5 1

) 2 3

) 4 5

25

1

1 2 3 x b x

1 3, 6 x x , c

0

1 0 1 a 0 c

1 1 , 0 b c , 0

0

a c e b 1 c

0 d , 0 1 b . a

) 1 1a

) 1 10

) b 0a

) x x 6.

) a b

) c 1

x x+a a x+a a 1 3 x 6 a a x+a 1 1 4+ x = 0. =0, x +1 x+2 2 x+2 3 x +1 = 0 , x+3

)

a a 3+ x

) x + 3 1

) 7.

5 6

4 2 1 2 a p c 3 0 =0, 3 p+c c+a =0, a+ p a1 b1 x a2 b2 x a3 b3 x c1 c3 0 xa 0 x+c bc ca = (b a )(c a)(c b) , ab c1 c2 , c3 x p xc = 0, 0

) 1 41

) x + a x+ p1 a b c b1 b2 b3

) 1 1

) 1 1a1 a3 c2 = 2 x a2

a1 + b1 x

) a2 + b2 x a3 + b3 x1 a b c a b c bc ab

1 1

a b c

a

2

1

a b c

a2 b 2 = (b a)(c a )(c b) , c2

) 1 11

ca = 1

b2 , c2 1 1 a b c

) 11 a2 b2 . c2

a3 c3

) 11

b3 = ( a + b + c ) 1

8.

, :2 x + 3 y + 5 z = 10 ) 3x + 7 y + 4 z = 3 x + 2 y + 2z = 3 5 x 6 y + 4 z = 3 ) 3x 3 y + 2 z = 2 4 x 5 y + 2 z = 1

26

4 x 3 y + 2 z + 4 = 0 ) 6 x 2 y + 3z + 1 = 0 5 x 3 y + 2 z + 3 = 0

5 x + 2 y + 3 z + 2 = 0 ) 2 x 2 y + 5 z = 0 3x + 4 y + 2 z + 10 = 0

x y +2 = 0 a b 2y ) b + 3cz 1 = 0 x z a + c = 0, abc 0 x + ay = 3 ) ax + z = 2 y + az = 1, a 1

2ax 3by + cz = 0 ) 3ax 6by + 5cz = 2abc 5ax 4by + 2cz = 3abc, abc 0

9.

, :2 x 3 y + z = 2 ) 3x 5 y + 5 z = 3 5 x 8 y + 6 z = 5 5 x 6 y + z = 4 ) 3x 5 y 2 z = 3 2 x y + 3z = 5 2ax 23 y + 29 z = 4 ) 7 x + ay + 4 z = 7 5 x + 2 y + az = 5 ax + 4 y + z = 0 ) 2 y + 3 z 1 = 0 3 x bz + 2 = 0 4 x + 3 y + 2 z = 1 ) x + 3 y + 5 z = 1 3x + 6 y + 9 z = 2 2 x y + 3z = 4 ) 3x 2 y + 2 z = 3 5 x 4 y = 2 ax 3 y + 5 z = 4 ) x ay + 3z = 2 9 x 7 y + 8az = 0 ax + 2 z = 2 ) 5 x + 2 y = 1 x 2 y + bz = 3

10. , +1 1 . 11. , +1 0. 12. :a x b b + y y2 2 2

c c + z z

a = (a 2 + b 2 + c 2 )( x 2 + y 2 + z 2 ) (ax + by + cz )2 . x c2 a c +a c = 11 2 3 c4 a3c1 + a4 c3 a1c2 + a2 c4 . a3c2 + a4 c4

13. a1 a3 a2 c1 i a4 c3

27

a3 = a2 , a4 = a1 , c3 = a2 , c4 = c1

2 2 2 2 (a1 + a2 )(c1 + c2 ) = (a1c1 a2 c2 )2 + (a2 c1 + a1c2 )2 .

28

II 1. 1.1. . () , .. , . , . , , Ox, Oy Oz , . , , . 1.2. M M , .. M Ox, Oy Oz . M x , M y M z x, y z

M x , M y M z Ox, Oy Oz , , .. x = OM x , y = OM y , z = OM z .. x , y - z - M . M ( x, y , z ) . 1.3. , , , ( x, y, z ) , (?). , ( x, y, z ) , x, y z . x, y z , Ox M x x , Oy M y y Oz M z

z . , M x Ox , M y Oy M z - Oz . M x, y z .

29

1.4. M Ox . M Oxy M xy .

M xy Ox M x . , M xy Oy , M y , . 1. , M Oxz Oyz , M xz M yz . , M x , M z M y , M z , .

M , M x , M y , M z , M xy , M xz , M yz O , M , . , M x, y, z , .. M ( x, y, z )

M x ( x,0,0), M y (0, y,0), M z (0,0, z ), M xy ( x, y,0), M xz ( x,0, z ), M yz (0, y, z ) .1.5. . Oxz y = 0 , Oxy z = 0 , Ox y = 0, z = 0 .

, Oy x = 0, z = 0 Oz x = 0, y = 0 .1.6. . Oxy, Oxz , Oyz , . M ( x, y, z ) ,

30

x > 0, y > 0, z > 0 , x < 0, y > 0, z > 0 , x < 0, y < 0, z > 0 , x > 0, y < 0, z > 0 , x > 0, y > 0, z < 0 ,

M M M M M

, , , , ,

-

x < 0, y > 0, z < 0 , M , x < 0, y < 0, z < 0 , M , x > 0, y < 0, z < 0 , M ,

2. . 2.1. . .

, , , , , .2.2. . , .

( A, B ) . ( A ) , ( B ) . AB = a , . 2. , : ) , .. , | a |=| AB |= AB , ) ) .2.3. . A B , AB . AB = o .

, , .. | o |= 0 .2.4. . AB CD AB CD ( ).

.2.5. . A, B C -

AB AC . !

31

. A, B C .

p , , AB AC p , .. . , AB AC . AB AC . , ( A ), , .. A, B C . 2.6. . AB CD , AB CD , ( ), AB CD (. 3 )). AB CD MN AB CD , AB CD (. 3 )). , , . 2.7. . , , ().

a = AB , BA a a .2.8. . a P . PQ

P , a . , , , .. . .2.9. . a = AB l . A ' B ' A B l , , a ' = A ' B ' a l . al = A ' B ' , a '

a l l , 32

, a l . , a = o , al = 0 .2.10. . e u , . 4, a ' = au e . (1)

2.11. . a l .

al =| a | cos , = ( a, l ) .

(2)

. , l . :

) l , .. [0, ] , 2 a ' = OA ' l (. 5). al = prl a = + | OA ' |=| OA | cos =| a | cos . ) l , .. ( , ] , 2 a ' = OA ' l (. 6). al = prl a = | OA ' |= | OA | cos( ) =| a | cos . 2.12. . :

) , , ) , , ) , .

33

2.13. . AB = CD l ,

prl AB = prl CD . 2.14. . 2.13 . , l , (. 7).

3. 3.1. . a x a Ox , y - a Oy z -

a Oz . 2.13 , a , . , b , a , a = b . , a b A B , . a b x Ox , A B , Ox x ( ). , A B Oy Oz . , , A B . b = OB = OA = a . .. a . 3.2. . x, y, z a

Ox, Oy, Oz a . a = {x, y , z} .34

3.3. . A( x1 , y1 , z1 ) B ( x2 , y2 , z2 )

AB x = x2 x1 , y = y2 y1 , z = z2 z1 . (1). A B Ox Ax Bx , . Ax Bx Ox x1 x2 , . , Ax Bx = x2 x1 Ax Bx = x x = x2 x1 .

y = y2 y1 , z = z2 z1 . 3.4. . Oxyz

M ( x, y, z ) . r = OM - M . OM (1). , x2 = x, y2 = y, z2 = z x1 = 0, y1 = 0, z1 = 0 OM = {x, y , z} .3.5. a = {x, y , z} ,

(. 9). A a Ax , Ay , Az Ox, Oy, Oz , . , OA . ,

OA = OA x + OA y + OA z ., OA =| a |, OAx = x, OAy = y, OAz = z , -

2

2

2

2

| a |2 = x 2 + y 2 + z 2 , ..

| a |= x 2 + y 2 + z 2 . (2) , (2) a x, y, z .3.6. A( x1 , y1 , z1 ) B ( x2 , y2 , z2 ) . d A B . 3.3 35

AB = {x2 x1 , y2 y1 , z2 z1} (2)

d = AB = ( x2 x1 ) 2 + ( y2 y1 ) 2 + ( z2 z1 ) 2 .3.7. . ) a = {1, 2, 2}

(3)

| a |= 12 + 22 + ( 2) 2 = 3 . ) AB, A(10, 20,0) B (30, 50, 40)

d = AB = (30 10)2 + (50 + 20) 2 + (40 0) 2 = 10 41 .

4. 4.1. . , ,

a . cos ,cos ,cos

a .4.2. a = {x0 , y0 , z0 } , (. 10):

cos =

x0 , |a|

cos =

y0 , |a|

cos =

z0 . |a|

, .. a | a | , cos ,cos ,cos x, y, z :

x =| a | cos , y =| a | cos , z =| a | cos ,

(1) (2)

| a |= x 2 + y 2 + z 2 .

4.3. . (1) (2) a , x, y, z . 36

cos =

x x + y +z2 2 2

, cos =

y x + y +z2 2 2

, cos =

z x + y2 + z22

.

(3)

, (3) , cos 2 + cos 2 + cos 2 = .. cos 2 + cos 2 + cos 2 = 1 . (4) , (4) , , , .x2 + y 2 + z 2 x2 + y 2 + z 2

5. 5.1. V . . a, b V b -

a . c a , b a b (. 11). c = a + b .5.2. . .

a + b b a . , a + b = o .5.3. a b b a (. 12). 5.1 c a b a b , .. c = a + b . . 12 ABCD . a = AB = DC , b = AD = BC c = AC . , b + a = AD + DC = AC = c , .. a + b = b + a .

, | a + b || a | + | b | || a | | b ||| a | + | b | .37

, a b , : ) a b , a + b | a | + | b | , a b. ) a b , a + b || a | | b || , a b . a V b a , .. b = a . 5.2 , a V a + (a) = o . , a, b, c V . a b a + b . , a + b c (a + b) + c . a + (b + c) , .. a b + c . ( a + b ) + c = a + (b + c ) . , a, b, c , b a , c b . , O a , A a , B b C c (. 13). ,

(a + b) + c = (OA + AB ) + BC = OB + BC = OC a + (b + c) = OA + ( AB + BC ) = OA + AC = OC .. (a + b) + c = a + (b + c) .5.4. 5.1 5.3 :

S i) a, b V , a + b V , S ii) a, b, c V (a + b) + c = a + (b + c) ,

38

S iii) o V a V a + o = o + a = a , S iv) a V a V a + ( a ) = ( a ) + a = o ,

S v) a, b V a + b = b + a .

6. 6.1. . a, b V . c

a b , c = a b , a = b + c .6.2. a b O A B , (. 14). c = b a . A . B , c a b . , b a = AB , . . a b . b a a b . 6.3. . a, b V .

a b = a + ( b ) ..

(1)

(a + (b)) + b = a + ((b) + b) = a + o = a . , a + (b) b a , 6.1 (1). 6.4. . a, b V . 6.2

a b a b . . , c = a b d = a b , 6.1 a = b + c a = b + d , .. b + c = b + d . , (b + c) + (b) = (b + d ) + (b) , c + (b + (b)) = d + (b + (b)) ,

c+o=d +o, c=d.

39

7. 7.1. . a V R . b a , | |i| a | a

> 0 , a < 0 , a . b = a .7.2. . a o . a 0 = 1 a |a|

a . 7.3. . a o . a 0 a | a 0 |=| 1 a |= 1 | a |= 1 ,|a| |a|

.. 1. a 0 a . , a a 0 | a | a 0 =| a | 1 a = a .|a|

7.4. 2.4 . . . a, b V b = ka

(1)

k R . . a b , b = o , b = 0ia . , | 0ia |= 0i| a |= 0 , 0ia = o = b . a b ( a o b o ) e e ' , (. 15). a =| a | e , b =| b | e ' .

(2) (3)

e = e ' ,

40

a b , a b . (2) (3) b =| b | e ' = | b | e = k = |b| |a| |b| |b| (| a | e) = a , |a| |a|

(1).

, (1), a b 7.1.

7.5. i) a V , R a V , ii) 1a = a , a V , iii) ( a ) = ( ) a , a V , , R , iv) ( + )a = a + a , a V , , R , v) (a + b) = a + b , a, b V , R .7.6. V , S i) S v) M i) M v). , .

8. 8.1. . a i V , i R, i = 1, 2,..., n .

a = i a ii =1

n

a i , i = 1, 2,..., n i , i = 1, 2,..., n .

8.2. . A B . C AB AC : CB = (. 16), AC CB AC = CB .(1)41

(1) -. , A, B, C O () OA, OB, OC , O

AB , AC CB AC = OC OA CB = OB OC . (1), OC OA = (OB OC ) OC + OC = OA + OB (1 + )OC = OA + OB+ OC = OA1+ OB .

(2) , OC C AB OA OB .

8.3. . a i , i = 1, 2,..., n i , i = 1, 2,..., n , , i =1

i a i = o .

n

8.4. . ) a i , i = 1, 2,..., n , a i , i = 1, 2,..., n . , a 1 = o , n

1ia 1 + 0ia i = oi =2

o, a i , i = 2,..., n . ) n i =1

a

a i , i = 1, 2,..., n . a = i a i , i , i = 1, 2,..., n ,

(1) a + i a i = oi =1

n

a , a i , i = 1, 2,..., n .42

) , C AB AC = CB O AB , (2), .. (1 + )OC OA OB = o , OC , OA, OB .

8.5. . a i , i = 1, 2,..., n , i a i = o i = 0, i = 1, 2,..., n .i =1 n

8.6. . a b . . a b . x y , ,

xa + yb = o . y 0 . b = x a , .. a b . y , a b , k a = kb , .. 1ia kb = o , a b .

8.7. . a b , 8.6 . , a b xa = yb x = y = 0 . 8.8. . a, b, c V , . 8.9. . a, b, c . . a, b, c , O (. 17). , , a b . c c a

c b , a b , . c = c a + c b = k a + mb , k , m R .43

, a b ,

a = kb = kb + 0c.. a b c . , a = OA, b = OB, c = OC ,

c = k a + mb , c a b , .. a, b, c .

9. 9.1. . a i , i = 1, 2,..., n u u , ..

pru ( a 1 + a 2 + ... + a n ) = pru a 1 + pru a 2 + ... + pru a n .. a i , i = 1, 2,..., n ,

.. a 1 a 2 , a 2 a 3 ., a n a n 1 (. 18). a 1 O , A1 , a 2 A2 .

a 1 + a 2 + ... + a n = OA n .

(1)

O, A1 ,..., An u

44

' ' ' O ', A1 , A2 ,..., An , . ' ' ' ' ' O ' A1 = pru a 1 , A1 A2 = pru a 2 , ..., An1 An = pru a n .

(2)

, (1) ' pru ( a 1 + a 2 + ... + a n ) = O ' An

(3)

' ' ' , O ', A1 , A2 ,..., An u ' ' ' ' ' ' O ' An = O ' A1 + A1 A2 + ... + An1 An

(4)

, (2), (3) (4) pru ( a 1 + a 2 + ... + a n ) = pru a 1 + pru a 2 + ... + pru a n . 9.2. . a V R u

pru a = pru a .. a O u A . a O B . , a = OA a = OB .

O, A, B , .. v . O, A, B O ', A ', B ' , (. 19).

O'B' O ' A'

= OB .OA

(5)

OB OA ( v ) , O ' B ' O ' A ' ( u ) . OB OA ( < 0 ), O ' B ' O ' A ' .

45

, O '' B '' OB , O A OA (5) O'B' O ' A'

= OB . OA

, a = OB a = OA , OB = , O '' B '' = , .. OA O A O ' B ' = iO ' A ' . , pru a = pru a . 9.3. .

a = {x1 , y1 , z1}, b = {x2 , y2 , z2 } c = a + b . 9.1

prOx c = prOx a + prOx b = x1 + x2 , prOy c = prOy a + prOy b = y1 + y2 , prOz c = prOz a + prOz b = z1 + z2 , a + b = c = {x1 + x2 , y1 + y2 , z1 + z2 } , .. {x1 , y1 , z1} + {x2 , y2 , z2 } = {x1 + x2 , y1 + y2 , z1 + z2 } .9.4. a = {x1 , y1 , z1} , R d = a . 9.2

(6)

prOx d = prOx a = x1 , prOy d = prOy a = y1 , prOz d = prOz a = z1 ,

a = d = { x1 , y1 , z1} ,..

{x1 , y1 , z1} = { x1 , y1 , z1} .

(7)

9.5. 8.2 . A B O , C AB -+ OC = OA1+ OB .

O A( x1 , y1 , z1 ) B ( x2 , y2 , z2 ) . OA = {x1 , y1 , z1} OB = {x2 , y2 , z2 } ,

46

+ OC = OA1+ OB = { 11+ 2 , 11+ 2 , 11+ 2 } ,

x + x

y + y

z + z

.. C ( 11+ 2 , 11+ 2 , 11+ 2 ) .9.6. . ) a = {2, 2, 4} b = {1, 2,3}

x + x

y + y

z + z

a + b = {2 + 1, 2 + 2, 4 + 3} = {3,0,7}, a b = {2 1, 2 2, 4 3} = {1, 4,1},2a = {2i2, 2i(2), 2i 4} = {4, 4,8}. , | a |= 22 + (2) 2 + 42 = 24 = 2 6 | b |= 12 + 22 + 32 = 14 | a + b |= 32 + 02 + 7 2 = 58 < 14 + 2 6 =| a | + | b | | a b |= 12 + (4)2 + 12 = 18 = 3 2 < 2 6 14 =|| a | | b || .) ABC : A(2,0,3); B (5, 1,3); C (1, 2,0) .

A1 BC = 1:1 5+1 , 1+ 2 , 3+ 0 ) , .. A (3, 1 , 3 ) . , T A1 ( 1+1 1+1 1+1 1 2 20+ 2i 1 3+ 2i 3

(. 20),

+ 2i AA1 = 2 :1 , T ( 21+ 23 , 1+ 2 2 , 1+ 2 2 ) , .. T ( 8 , 1 , 6 ) . 3 3 3

10. 10.1. i, j , k :

1) i Ox , j - Oy k - Oz ; 2) i, j , k ; 3) i, j , k , .. | i |=| j |=| k |= 1 . i, j , k .47

a , A . A , Oz B Oxy . B Ox Oy Oy Ox Ay Ax , . , A OB Oz Az (. 21).

a = OB + OA z , OB = OA x + OA y , a = OA x + OA y + OA z . (1)

, OA x i , OA x = xi x R . , OA y = y j OA z = zk , y , z R . (1) a = xi + y j + zk . .. i, j , k . 10.2. . (2) a a i, j , k . x, y, z

(2)

, xi, y j , zk a i, j , k .10.3. OA x = xi i

Ox x = OAx . , OAx a = OA Ox , x = prOx a . , y = prOy a z = prOz a . , .

48

. a i, j , k , .. (2), x, y, z a , .. . 10.4. i, j , k . ,

m, n, p .. m, n, p . ,

a m, n, p : a =m + n + p . (3). i, j , k . .

11. 11.1. . a b .

| a |i pra b =| b |i prb a

(1)

. a b , pra b = prb a = 0 , , -

(1). a b , prb a =| a |, pra b =| b |, , prb a = | a |, pra b = | b |, | a |i pra b = | a |i| b |=| b |i prb a . a b . O , .. OA = a , OB = b (. 22 23). OA ' A OB ' B

49

O , . OA ' : OB ' = OA : OB , .. OAiOB ' = OA 'iOB , | a |i| pra b |=| b |i| prb a | . , prb a pra b ( . 22 , . 23 ), (1). 11.2. . a, b, c

a + b + c = o . !. a, b, c -

, .. ABC a = BC , b = CA, c = AB . a + b + c = BC + CA + AB = BC + (CA + AB ) == BC + CB = o . , a + b + c = o . A B , c = AB , C , a = BC ( ). CA : CA = CB + BA = BC AB = a c = (a + c) . a + b + c = o b = (a + c) , b = CA , .. a, b, c ABC . 11.3. . a b . x a b . . m n . m = OM n = ON OMPN , m + n = OP . , m + n OMPN , O .

, m + n m n m n | m |=| n | . , , , a b . | b | a | a | b | a |i| b | , x =| b | a + | a | b . 11.4. . M N AB . OM ON a = OA b = OB , O AB .

50

. X AB = m : n ,

8.2 OX = nOA+ mOB . m+ n M AB 1: 2 , N AB 2 :1 . :+ OM = 2OA3 OB ON = OA+32OB .

11.5. . -

.. a, b, c, d .

, a, b c , c = xa + yb x y , xa + yb + (1)c + 0id = o , .. a, b, c, d .

, a, b, c, d . O ,

a = OA, b = OB, c = OC , d = OD(. 24). D OC AOB E . a = OA, b = OB, OE , OE = xa + yb ,

x, y R . ED c , ED = zc z R . , d = OD = OE + ED = xa + yb + zc..

xa + yb + zc + (1)d = o ,

51

a, b, c, d .

11.6. . E AB ABCD M = AC DE . AM : MC DM : ME . .

AB = a, AD = b

(. 25). AM MC , x R AM = x AC . AC = a + b , AM = x(a + b) . ,

y R DM = yDE = y (b + 1 a) . 2 AD + DM = AM ,

b + y ( b + 1 a ) = x ( a + b ) 2.. ( x 1 y ) a + ( x + y 1)b = o . 2 (1) a b , (1) x 1 y = 0 2

x + y 1 = 0 , x = 1 , y = 2 . 3 3, AM = 1 AC = 1 ( AM + MC ) , .. 2 AM = MC , 3 3 AM : MC = 1: 2 . , DM = 2 DE DM : ME = 2 :1 . 3

11.7. . M N AC BC ABC , MN AB . M N AC BC . . MN AB , MN AB , .. R , MN = AB . M AC m : n , N BC p : q . :MN = MC + CN = mn n AC + p + q CB . +p

,

AB = mn n AC + +

p CB p+q p CB p+q

( AC + CB ) = mn n AC + +p

( mn n ) AC + ( p + q )CB = o. +

52

p , AC CB , = mn n , = p + q . +

= p + q m : n = p : q .. M N AC BC .

n m+ n

p

11.8. . . ! . ABCD - A, B, C , D r 1 , r 2 , r 3 , r 4 , . M (r M ) N ( r N ) AB CD , r M = 1 (r 1 + r 2 ), r N = 1 (r 3 + r 4 ) . 2 2 , , P (r P ) , Q( r Q ) , S (r S ) T ( r T ) AC , BD, AD BC , r P = 1 ( r 1 + r 3 ), r Q = 1 (r 2 + r 4 ), r S = 1 (r 1 + r 4 ), r T = 1 (r 2 + r 3 ) . 2 2 2 2 , : r M + r N = r P + r Q = r S + r T = 1 (r 1 + r 2 + r 3 + r 4 ) 2 MN , PQ ST , (?).

11.9. . n . n . . S1 ( r 1 ), S2 (r 2 ), ..., Sn (r n ) A1 A2 ... An n , S1 A1 A2 , S2 ' ' A2 A3 . Sn An A1 . A1 (r1 ), A2 (r 2 ),

' ..., An (r n ) , ' ' r1 + r 2 = 2r 1 , ' ' r 2 + r 3 = 2r 2 ,

.......................' ' r n 1 + r n = 2r n 1 , ' ' r n + r1 = 2r n .

53

1 , . ' r1 = r 1 r 2 + r 3 ... r n 1 + r n . ' , A1 (r1 ) , -

n .

11.10. . d = {2, 4, 2} a = {1, 2,5}, b = {1,6,3}, c = {0,0, 2} .

. x, y, z d = xa + yb + zc . : {2, 4, 2} = x{1, 2,5} + y{1,6,3} + z{0,0, 2} = {x y , 2 x + 6 y,5 x + 3 y + 2 z} ..

x y = 2 2 x + 6 y = 4 5 x + 3 y + 2 z = 2. , x = 2, y = 0, z = 4 . d = 2a + 0ib + (4)c .

11.11. . ) a = {a1 , a2 , a3}, b = {b1 , b2 , b3}, c = {c1 , c2 , c3} a1 b1 c1 . a2 b2 c2 a3 b3 = 0 . c3 (1)

) a = {3,0, 2}, b = {2,1, 4}, c = {11, 2, 2} . ) a, b, c . x, y, z , xa + yb + zc = o , ..

54

a1x + b1 y + c1 z = 0 a2 x + b2 y + c2 z = 0 a x + b y + c z = 0 3 3 3 . , (1). .

) 3 2 11 0 1 2 2 4 = 6 8 22 + 24 = 0 2

a, b, c .

11.12. . ) A(a1 , a2 , a3 ), B (b1 , b2 , b3 ), C (c1 , c2 , c3 ) D( d1 , d 2 , d3 ) b1 a1 c1 a1 d1 a1 b2 a2 c2 a2 d 2 a2 b3 a3 c3 a3 = 0 . d3 a3 (1)

) A(2,3,1), B (3,1, 4), C (2,1,5), D (0,0,9) ? . ) A, B, C , D AB, AC , AD , , 11.11 (1). ) 32 22 02 1 3 1 3 03 4 1 1 5 1 = 0 9 1 2 2 2 3 3 4 = 16 + 16 12 + 12 = 0 , 8

A, B, C , D .

11.13. . AB CD , A(6,0,1), B (1,3, 2), C (5,1, 3), D (6,1,3) . . AB AB = {7,3,1} , CD CD = {1,0,6} . AB CD , , AB CD . , AB CD . , AB , AC CD . AC = {1,1, 4} , 55

7 1 1

3 0 1

1 6 = 18 + 1 + 42 + 12 = 37 0 . 4

11.11, AB , AC CD . , AB CD .

12. 12.1. . a b . | a |i| b | cos , = ( a, b) a b . ab =| a |i| b | cos , = ( a, b ) . (1)

12.2. . , | b | cos = pra b , ab =| a | pra b . , | a | cos = prb a ab =| b | prb a . (3) (2)

, .

12.3. . a b ab = ba . (4)

. cos = cos( ) , = ( a, b) ab =| a |i| b | cos =| b |i| a | cos( ) = ba .

12.4. . a, b c (a + b)c = ac + bc . (5)

. (3) 9.1 (a + b)c =| c | pr c (a + b) =| c | ( pr c a + pr c b) =| c | pr c a + | c | pr c b = ac + bc . 56

12.5. . a b R ( a)b = (ab) .

. (3) 9.2 ( a)b =| b | prb ( a) =| b | prb a = (| b | prb a ) = (ab) .

12.6. . a, b V , R ( a)( b) = ( )(ab) .

. 12.3 12.5 ( a)( b) = (a ( b)) = (( b)a) = (( (ba )) = ( )(ba) = ( )(ab) .

12.7. . a, b, c V , R ( a + b)c = (ac) + (bc) .

. 12.4 12.5 ( a + b)c = ( a)c + ( b)c = (ac) + (bc) .

12.8. . , . , , (ab)c ab c , .. c .

12.9. . a, b V ab = 0 .

. a b , ..

(a, b) = . (1) 2

ab =| a |i| b | cos =| a |i| b |i0 = 0 . 2 , a = o b = o , a b . a o , b o ab = 0 . (1) cos (a, b) = ab = 0 = 0 ,|a|i|b| |a|i|b|

(a, b) = , .. a b . 2

57

12.11. . n p q , | p |= 2, | q |= 3, n . ( p, q) = , n p = 7, nq = 3 . 3

. n = x p + yq, x, y R . n p = 7 7 = n p = ( x p + yq ) p = x p p + y pq = x | p |2 + y | p |i| q | cos = 22 x + 2i3i 1 y , 3 2 .. 7 = 4 x + 3 y . nq = 3 3 = nq = ( x p + yq)q = x pq + yqq = x | p |i| q | cos + y | q |2 = 2i3i 1 x + 32 y , 3 2 .. 3 = 3x + 9 y .

4 x + 3 y = 7 3x + 9 y = 3 x = 2, y = 1 . , n = 2 p 1 q . 3 3 , | n |2 = nn = (2 p 1 q)(2 p 1 q ) = 4 p p 4 pq + 1 qq 3 3 3 94 1 = 4 | p |2 4 | p |i| q | cos + 1 | q |2 = 4i22 3 i2i3i 1 + 9 i32 = 13 3 3 9 2 | n |= 13 .

12.12. . a b a + 3b 7 a 5b a 4b 7 a 2b . . 12.9 ( a + 3b)(7 a 5b) = 0 ( a 4b)(7 a 2b) = 0 7 + 21b 5b 15bb = 0 7 28b 2b + 8bb = 0 ..7 | |2 +16 | |i| b | cos 15 | b |2 = 0 7 | |2 30 | |i| b | cos + 8 | b |2 = 0

58

cos =

15|b|2 7||2 16||i|b|

(6)

7 | |2 30 | |i| b |

15|b|2 7||2 16||i|b|

+ 8 | b |2 = 0,

56 | |2 225 | b |2 +105 | |2 +64 | b |2 = 0, 161| a |2 = 161| b |2 , | a |=| b | . , (6) cos = = arccos 1 = . 2 315|b|2 7|b|2 16|b|i|b|

=1 2

13. 13.1. . a = {x1 , y1 , z1} b = {x2 , y2 , z2 } ,

ab = x1 x2 + y1 y2 + z1 z2 .. i, j , k ii = 1, i j = 0, ik = 0 ji = 0, j j = 1, jk = 0 ki = 0, k j = 0, k k = 1.

(1)

(2)

a b : a = x1 i + y1 j + z1 k b = x2 i + y2 j + z2 k .

(3)

(3), ,

ab = x1x2 ii + x1 y2 i j + x1z2 ik + y1x2 ji + y1 y2 j j + y1z2 jk + z1x2 ki + z1 y2 k j + z1z2 kk , (2) (1). 13.2. . a = {x1 , y1 , z1} b = {x2 , y2 , z2 } 59

x1x2 + y1 y2 + z1 z2 = 0 .

(4)

. 12.9 a b ab = 0 . 13.1 a b (4). 13.3. . a = {x1 , y1 , z1}

b = {x2 , y2 , z2 } cos =x1x2 + y1 y2 + z1z2 2 2 2 2 2 2 x1 + y1 + z1 x2 + y2 + z2

.

(5)

. ab =| a |i| b | cos -

cos = ab .|a|i|b|

(6)

, 13.1 ab = x1 x2 + y1 y2 + z1 z2 , 4.22 2 2 2 2 2 | a |= x1 + y1 + z1 , | b |= x2 + y2 + z2 . (6) (5).

13.4. . u , , , s = {x, y, z} u pru s = x cos + y cos + z cos . (7) . u e . es =| e | pre s | e |= 1 , pre s = pru s pru s = es . ,

u , u , .. , , . prOx e =| e | cos , prOy e =| e | cos , prOz e =| e | cos , | e |= 1 , e = {cos ,cos ,cos } . pru s = es = {cos ,cos ,cos }i{x, y, z} = x cos + y cos + z cos .. (7). 13.5. . x a = {x 2, 4,1} b = {1,3,5} . . a b ab = 0 , ..60

1( x 2) + 3i 4 + 1i5 = 0 x + 2 + 12 + 5 = 0 x = 19 . 13.6. . a b a = {1, 2, 2} b = {1,0,1} . . prb a = ab ab = 3 | b |= 2 |b|

prb a =

3 2

.

13.7. . A(1,1,1), B (2, 2,1) C (2,1, 2) . = BAC . . AB = {1,1,0} AC = {1,0,1} . 13.3 1i1+1i 0+ 0i1 cos = =1. 212 +12 + 02 12 + 02 +12

, = arccos 1 = . 2 313.8. . A(1,1,1) B (4,5, 3) . -

AB u , .. cos , cos , cos u . = = , cos = cos = cos

cos 2 + cos 2 + cos 2 = 1 3cos 2 = 1 , .. cos 2 = 1 3cos = cos = cos = 1 .3

AB = {3, 4, 4} (7) pru AB = 3i 1 + 4i 1 + (4)i 1 = 3 . 3 3 3

61

14. 14.1. . a, b, c V .

a, b, c , , .

a, b, c , , .14.2. . a, b, c : a , b, c b, a , c b, c , a c , b, a c, a, b a, c, b .

, , . i, j , k , . , a, b, c , .14.3. . a, b V . a

b c : ) | c |=| a |i| b | sin , = ( a, b) , ) c a b , ) a, b, c . c = [a, b] c = a b , (. 27).14.4. . a b . . a b . , =

( a, b) = 0

= (a, b) = , a b , . sin = 0 , | [a, b] |=| a |i| b | sin = 0 , .. [a, b] = o .

62

[a, b] = o . | [a, b] |= 0 , | a |i| b | sin = 0 . a o b o , sin = 0 , = 0 = , .. a b . a = o b = o , a b . 14.5. . a b O , [a, b] a b . . O a b OADB , (. 28), S . S =| a |i H =| a |i| b | sin =| [a, b] | .

(1)

14.6. . 14.5 a b

[a, b] = Se

(2)

S a b , e ) | e |= 1 , ) e a b , ) a, b, e .14.7. . a b

[a, b] = [b, a] .

(3)

. a b , [a, b] = o = [b, a] , (3).

a b . , ) ) 14.3 [a, b] [b, a] . , a , b [a, b] ,

63

b , a [b, a ] [a, b] [b, a] (. 29), (3). 14.8. . a, b V R

[ a , b ] = [ a , b ] , [ a, b] = [ a, b] .

(4) (5)

. = 0 a b , (4). , (4) . 0 a b . 14.3 [a, b] | a |i| b | sin , = ( a, b) . , [a, b] | |i| a |i| b | sin . , [ a, b] | |i| a |i| b | sin , = [a, b] . , ) 14.3 [a, b] [ a, b] a b , [a, b] [ a, b] . > 0 . a a , , , [ a, b] [a, b] . , > 0 [a, b] [a, b] , [ a, b] [a, b] . , [ a, b] = [a, b] , .. (4). < 0 . a a , , , [ a, b] [a, b] . , < 0 [a, b] [a, b] , .. [ a, b] . , [ a, b] = [a, b] , .. (4). , (4) 14.7 64

( a, b) . , = > 0

= < 0 , [ a, b]

[a, b] = [ b, a] = [b, a ] = [a, b] .

14.9. . a, b, c V [ a , b + c ] = [ a, b ] + [ a , c ] , [b + c, a] = [b, a] + [c, a] . (6) (7)

. a = o , (6). , a b c . a 0 , a 0 OB OC . OD = OB + OC , . 30.

OB* = [a 0 , OB ], OC* = [a 0 , OC ] OD* = [a 0 , OD] = [ a 0 , OB + OC ]. ) ) 14.3 ) | OB*|=| [a 0 , OB] |=| a 0 |i| OB | sin =| OB | 2 ) OB* a 0 , OB* OB . , OB* OB a 0 . 2 , ) 14.3 a 0 , OB OB* , .. a 0 (?). , OC OD a 0 2 OC * OD* . , OB * C * D * OBCD , OB * C * D * . OD* = OB* +OC * , ..

65

[a 0 , OD ] = [a 0 , OB] + [ a 0 , OC ] , (6).

(8)

a , OB OC a 0 a , .. a =| a | a 0 . (8) | a | [a, OD] = [ a, OB ] + [a, OC ] . a, b, c . a, b, c O . b, c b + c a . , O a B, C D , (. 31). [a, b] [a, OB ] . , a b a OB . , [a, b] [a, OB ] , a , b OB . , [a, b] [a, OB ] , [a, OB ] = [a, b] . [a, OC ] = [a, c] [a, OD] = [ a, b + c] . , (9) [ a , b + c ] = [ a, b ] + [ a , c ] , .. (6). , : [b + c, a] = [a, b + c] = [a, b] [a, c] = [b, a] + [c, a ] , .. (7).

(9)

66

14.10. . a b (. 32). a + b a b . :[a b, a + b] = [a, a] + [a, b] [b, a] [b, b] = o + [a, b] + [a, b] + o = 2[a, b] , | [a b, a + b] |= 2 | [a, b] | . , a + b a b a b .

14.11. . p = a + 2b q = a 3b , | a |= 5 , | b |= 3 (a, b) = . 6 . [ p, q] = [a + 2b, a 3b] = [a, a ] 3[a, b] + 2[b, a ] 6[b, b] = 5[b, a ]. , S =| 5[b, a] |= 5 | b |i| a | sin (b, a) = 5i5i3sin = 75 . 6 2

| p |2 = p p = (a + 2b)(a + 2b) = aa + 4ab + 4bb =| a |2 +4 | a |i| b | cos + 4 | b |2 = 52 + 4i5i3i 23 + 4i32 = 61 + 30 3, 6 .. | p |= 61 + 30 3. , H= S =| p| 75 2 61+30 3

.

67

15. 15.1. . a = {x1 , y1 , z1} b = {x2 , y2 , z2 } , y [ a, b] = { 1 y2 z1 x , 1 z2 x2 z1 x1 , z2 x2 y1 }. y2 (1)

. , , [i, i ] = [ j , j ] = [k , k ] = o . [i, j ] . , | [i, j ] |=| i |i| j | sin = 1 , .. [i, j ] 2 i j , k . , [i, j ] = k . [ j , k ] = i [k , i ] = j , [ j , i ] = k , [k , j ] = i [i, k ] = j , [i, i ] = o, [ j , i ] = k , [k , i ] = j , a b a = x1 i + y1 j + z1 k b = x2 i + y2 j + z2 k [a, b] = x1x2 [i, i ] + x1 y2 [i, j ] + x1 z2 [i, k ] + y1x2 [ j , i ] + y1 y2 [ j , j ] + y1 z2 [ j , k ] + z1 x2 [k , i ] + z1 y2 [k , j ] + z1 z2 [k , k ] (2)

[i, j ] = k , [ j , j ] = o, [ k , j ] = i ,

[i , k ] = j , [ j , k ] = i, [k , k ] = o. (2)

(3)

[a, b] = ( y1 z2 z1 y2 )i ( x1 z2 z1x2 ) j + ( x1 y2 y1x2 )k y = 1 y2 y ={ 1 y2 z1 i z2 x1 x2 z1 j+ z2 z1 x1 , z2 x2 x1 x2 y1 k y2 y1 } y2

z1 x , 1 z2 x2

.

15.2. . (1)

68

i [a, b] = x1 x2

j y1 y2

k z1 , z2

(4)

(4) . 15.3. . ABC : A(1,1,0), B (1,0,1) C (0,1,1) . . ABC AB AC (. 34). , AB = {0, 1,1}

AC = {1,0,1} ,

i [ AB, AC ] = 0 1,

j 1 0

k 1 = i j k = {1, 1, 1} . 1

P = 1 | [ AB, AC ] |= 1 ( 1) 2 + (1)2 + (1) 2 = 23 . 2 215.4. . M (2,3,5) A(1,1, 2)

a = {3,0, 4} .. a AM . P =| [a, AM ] | , H

a |[ a , AM ]| H=P= |a| |a|

M A a (. 35). , AM = {3, 2,3}

69

i [a, AM ] = 3 3,

j 0 2

k 4 = 8i 3 j 6k = {8, 3, 6} . 3

( 8)2 + ( 3) 2 + ( 6)2 |[ a , AM ]| H=P= = = 109 . 5 |a| |a| ( 3)2 + 02 + 42

16. 16.1. . a, b, c V .

(a, b, c) = [a, b]c a, b, c .

(1)

16.2. a, b, c a

, b c . , b, c, a , b , c a . 14.1 . , . , a, b, c , a , b, c b, c , a c, a, b b, a , c c , b, a a, c, b .

, , a , b, c b, c , a c , a , b , , b, a , c c , b, a a, c, b

.16.3. . [a, b]c

a, b, c , , a, b, c , a, b, c .

70

. , a b . S a b , e , 14.6.

[a, b] = Se .

[a, b]c = S (ec) = S | e | pre c = S i pre c . (1) , pre c = h , h , a, b, c a b . , V = Sh , [a, b]c = V . (2)

, a, b, c , e c a b , pre c = h , .. [a, b]c = V . , a, b, c , e c a b , pre c = h , .. [a, b]c = V . a b , [a, b] = o , [a, b]c = 0 = V , . 16.4. . a, b, c . . a, b, c . pre c = 0 e

14.6. 16.3 [a, b]c = S i pre c = S i0 = 0 a, b, c . , V a, b, c . , 16.3 [a, b]c = V 0 . 16.5. . a, b, c V

71

[a, b]c = a[b, c].

(3)

[a, b]c = [b, c]a . , 16.3 [a, b]c = V , [b, c]a = V .

(4)

(5)

, a, b, c b, c, a , 16.3 (5) . , [a, b]c = [b, c]a , (4) (6) (3). 16.6. . (3) [a, b]c (a, b, c) . , (3)

(6)

[a, b]c = a[b, c] , (a, b, c) .16.7. . a, b, c V

(a, b, c) = (b, c, a) = (c, a, b) . a, b, c ; b, c, a ; c, a, b , (7). 16.8. . a, b, c V

(7)

. , (a, b, c) = V , (b, c, a) = V , (c, a, b) = V . ,

(b, a, c) = (a, c, b) = (c, b, a) = (a, b, c) .

(8)

. , (b, a, c) = V , (a, c, b) = V , (c, b, a) = V -

b, a, c ; c, b, a ; a, c, b (b, a, c) = (a, c, b) = (c, b, a) . , (b, a, c) = [b, a]c = [a, b]c = ( a, b, c) . , (9) (10) (8). 16.9. . m1 : n1 , m2 : n2 m3 : n3 . .72

(9) (10)

. O1 O2 , O1 A1 , O1B1 , O1C1 O2 A2 , O2 B2 , O2C2 , .

O2 A2 = n1 O1 A1 , O2 B 2 = n 2 O1B1 , O2C 2 = n 3 O1C1 . 1 2 3 V1 V2 ,

m

m

m

V2 = (O2 A2 , O2 B 2 , O2C 2 ) = ( n1 O1 A1 , n 2 O1B1 , n 3 O1C1 )1 2 3

m

m

m

=mm m

m1m2 m3 mm m (O1 A1 , O1B1 , O1C1 ) = n1 n 2n 3 V1 n1n2 n3 1 2 3

.. V2 : V1 = n1 n 2n 3 . 1 2 316.10. . x

ax = , bx = , cx =

(11)

a, b, c , , , .. (11) , -

ax = , bx = , ( a b) x = 0 , ax = , cx = , .. ( a c) x = 0 . , x a b a c , [ a b, a c] = ( [a, b] + [b, c] + [c, a ]) . , x = t ( [a, b] + [b, c] + [c, a ]) , t R . (12) (12) c ( 0 ) cx = t ( (a, b, c) + (b, c, c) + (c, a, c)) (b, c, c) = (c, a, c) = 0 = t ( a, b, c) . , a, b, c , .. (a, b, c) 0 0 ,

73

t =

1 ( a ,b,c )

.

, (12) x=1 ( [ a, b] + [b, c ] + [c, a ]) . ( a ,b,c )

(13)

17. 17.1. . a, b, c a = {x1 , y1 , z1} , b = {x2 , y2 , z2 } c = {x3 , y3 , z3} , x1 (a, b, c) = x2 x3 y1 y2 y3 z1 z2 . z3

. (a, b, c) = [a, b]c . 15.1 y [ a, b] = { 1 y2 z1 x , 1 z2 x2 z1 x1 , z2 x2 y1 }. y2

c = {x3 , y3 , z3} y (a, b, c) = x3 1 y2 z1 x y3 1 z2 x2 z1 x + z3 1 z2 x2 x1 y1 = x2 y2 x3 y1 y2 y3 z1 z2 z3

.

17.2. . a = {x1 , y1 , z1} , b = {x2 , y2 , z2 } c = {x3 , y3 , z3} x1 x2 x3 y1 y2 y3 z1 z2 = 0 . z3

. 16.4 17.1. 17.3. . Oy A(3, 2,5), B(0, 4, 3) C (1,1, 2) .

74

. M (0, y,0) , Oy A, B, C AM , AB, AC . :AM = {3, y + 2, 5}, AB = {3,6, 8}, AC = {4,3, 3} 17.2 AM , AB, AC 3 3 4 y+2 6 3

5 8 = 0 , 3

.. y = 47 . , M (0, 47 ,0) . 23 23

17.4. . ABCD : A(2, 4,5) , B (1, 3, 4), C (5,5,1), D(1,1, 2) D . . AB, AC , AD . V = ( AB, AC , AD ) , B =| [ AB, AC ] | , (. 37). , H =V = B ( AB , AC , AD ) |[ AB, AC ]|

.

: AB = {3,1, 1}, AC = {3,9, 4}, AD = {1,5, 7} .

3 V = ( AB, AC , AD) = 3 1 i [ AB, AC ] = 3 3 j 1 9

1 9 5

1 4 = 189 + 4 15 9 60 + 21 = 214 84 = 130 7

k 1 = 4i 3 j 27 k 3k + 9i 12 j = {5, 15, 30} ,

4

.. B =| [ AB, AC ] |= 52 + (15) 2 + (30) 2 = 5 46 . ,

75

46 H =V = = 130 = 1323 . B |[ AB, AC ]| 5 46

( AB, AC , AD )

17.5. . a = {1,1,1}, b = {2, 1,1} c = {3,5, 2} . x ax = 3, bx = 0 , cx = 6 . (1)

. x = {x, y , z} . (1) x + y + z = 3 2 x y + z = 0 3x + 5 y 2 z = 6.: = 17, x = 3, y = 27, z = 21 , 3 27 21 x = x = 17 , y = y = 17 , z = z = 17 .

(2)

3 27 21 , x = {17 , 17 , 17 } .

(13) 16.10.

18. 18.1. . a, b c . b c , a [b, c] , [a,[b, c]] .

18.2. . a, b c [a,[b, c]] = (ac)b (ab)c . (1)

. a = {x1 , y1 , z1}, b = {x2 , y2 , z2 }, c = {x3 , y3 , z3} . :y [b, c] = { 2 y3 z2 x , 2 z3 x3 z2 x2 , z3 x3 y2 } = { y2 z3 z2 y3 , x2 z3 + z2 x3 , x2 y3 y2 x3} y3

76

[a,[b, c]] = {

y1 z2 x3 x2 z3 x1 y2 z3 z2 y3

z1 x1 , x2 y3 y2 x3 y2 z3 z2 y3 y1 } z2 x3 x2 z3

z1 , x2 y3 y2 x3

= { y1x2 y3 y1 y2 x3 z1 z2 x3 + z1 x2 z3 , x1 x2 y3 + x1 y2 x3 + z1 y2 z3 z1 z2 y3 , x1 z2 x3 x1x2 z3 y1 y2 z3 + y1 z2 y3} = {( y1 y3 + z1 z3 + x1 x3 ) x2 ( y1 y2 + z1 z2 + x1x2 ) x3 , ( y1 y3 + z1 z3 + x1x3 ) y2 ( y1 y2 + z1 z2 + x1x2 ) y3 , ( y1 y3 + z1 z3 + x1x3 ) z2 ( y1 y2 + z1 z2 + x1 x2 ) z3} = ( y1 y3 + z1 z3 + x1 x3 ){x2 , y2 , z2 } ( y1 y2 + z1z2 + x1 x2 ){x3 , y3 , z3} = (ac)b (ab)c .

18.3 . 18.2 [a,[b, c]] b c . , [[a, b], c]] = [c,[a, b]] 18.2 [[a, b], c]] b a . , [a,[b, c]] [[ a, b], c]] , a, b c .

18.4. . a b . x x + [ a, x ] = b . (2)

. (2) , , a [a, x] + [a,[a, x]] = [a, b] . , [a, x] = b x 18.2 [a,[a, x]] = ( ax) a ( aa) x , (3)

77

(3) b x + ( ax)a ( aa) x = [ a, b] . (4)

(2) a ax = ab (?) (4) b x + ( ab)a | a |2 x = [a, b] b + ( ab)a [a, b] = (1+ | a |2 ) x x= .. x =b + ( ab ) a [ a ,b ]

(1+|a|2 )

,

1 (b + ( ab) a [ a, b]) . (1+|a|2 )

18.5. . x xa = , [ x, b ] = c ,

a, b, c , ?

. x [ x, b] = c , b c . [ x, b] = c a [a,[ x, b]] = [a, c] (ab) x (ax)b = [a, c] (ab) x b = [a, c] (ab) x = b + [a, c]. a b , ab 0 , x = 1 ( b + [a, c]) .ab

(5)

: b c , x , b c , a b , x (5), b , c a , b , b + [a, c] o , ,

-

78

-

b , c a , b , b + [a, c] = o , x .

19. 19.1. : , O i, j , k , O ' i ', j ', k ' .

x, y, z M x ', y ' z ' M . x, y, z OM i, j , k x ', y ' z ' OM ' i ', j ', k ' , .. OM = xi + y j + zk (1)OM ' = x ' i '+ y ' j '+ z ' k ' .

(2)

a, b, c O ' . OO ' = ai + b j + ck .

(3)

i, j , k , im , i = 1, 2,3; m = 1, 2,3 i ' = 11 i + 12 j + 13 k j ' = 21 i + 22 j + 23 k k ' = 31 i + 32 j + 33 k . OM = OO '+ O ' M . i, j , k (5) (4) (2), (5) OM , OO ', O ' M

(4)

xi + y j + zk = (a + 11x '+ 21 y '+ 31z ')i + (b + 12 x '+ 22 y '+ 32 z ') j + (c + 13 x '+ 23 y '+ 33 z ')k .

(6)

79

10.3 x = a + 11 x '+ 21 y '+ 31 z ' y = b + 12 x '+ 22 y '+ 32 z ' z = c + x '+ y '+ z ' 13 23 33 (7)

. .

. x, y, z x ', y ' z ' . 19.2. im , i = 1, 2,3; m = 1, 2,3 (4). (4) i, j , k i, j , k , i ', j ', k ' .

11 = cos (i, i '), 21 = cos (i, j '), 31 = cos (i, k '),

12 = cos ( j , i '), 22 = cos ( j , j '), 32 = cos ( j , k '),

13 = cos (k , i '), 23 = cos (k , j '), 33 = cos (k , k ').

20. 1. 2. 3. 4. ABCD O OA + OC = OB + OD . ABCDEF . a = AB b = AF BC , CD, ED FE . OA1 , OA 2 ,..., OA n A1 A2 ... An n O . ) 3(a 2b) 2(3a b) + 4(a + b) , ) 2(a 3b + 2c) + 4b 2(a b + 3c) . 5. ( x y ): 2 x + 3 y = a + b ) x + 2 y = a b 2 x y = b 2a ) 4 x 2 y = 2a.

80

6. 7. 8. 9.

a = p 2q, b = q 3 p . c , a, b, c . M , N , P BC , CA, AB , ABC . AM , BN , CP . A, B, C D , M N AB CD . 2MN = AC + BD = AD + BC . AB CD (O, r ) S . SA + SB + SC + SD = 2SO .

10. M , N P AB . OM , ON OP a = OA b = OB , O AB . 11. M N AD BC ABCD . MN AB DC .

12. a, b, c . p = x1 a + y1 b + z1 c q = x2 a + y2 b + z2 c r = x3 a + y3 b + z3 c x1 y1 z1 x2 y2 z2 = 0 . x3 y3 z3

13. a, b, c . p, q, r : ) p = 3a + 2b c, q = 7a + 2b 3c, r = 10a + 4b 4c , ) p = a + 2b + 2c, q = 2a b + 2c, r = 2a + 2b c . 14. , a, b, c x, y, z p = xa yb, q = zb xc, r = yc za . 15. ABC s = AM A . s b = AC c = AB . 16. V ABC . AV b = AC c = AB .

17. A B BC CA ABC , , P = AA BB . AP b = AC c = AB .

81

18. AB ABCD E , AE : ED = k . P = AC BE , AC : AP BP : BE .

19. E AB , F BC ABCD S = AF DE , T = EF BD . ES : SD, AS : SF , BT : TD, ET : TF .

20. E F AB BC ABCD , S = AF DE , T = EF BD . ES : SD, AS : SF , BT : TD, ET : TF . 21. AA1 , BB2 ABC P = AA1 BB2 . P AA1 BB2 . 22. A ', B ', C ' BC , CA, AB ABC . ABC A ' B ' C ' . 23. A ', B ', C ' BC , CA, AB ABC , T A '', B '', C '' AT , BT , CT . A ' B ' C ' A '' B '' C '' T . 24. M (m), N (n), P ( p) BC , CA, AB ABC . . 25. d a, b, c : ) a = {1,3, 0}, b = {5,10, 0}, c = {4, 2, 6}, d = {10,11, 6} , ) a = {1,3,5}, b = {0,5, 4}, c = {7, 8, 4}, d = {5,19,10} . 26. a, b, c : ) a = {1, 0, 7}, b = {1, 2, 4}, c = {3, 2,1} , ) a = {5,1, 4}, b = {3, 5, 2}, c = {1, 13, 2} . 27. a = {6, 4, 2}, b = {9, 6,3}, c = {3, 6,3} . x, y, z , xa, yb, zc . 28. a = {1,5,3}, b = {6, 4, 2}, c = {0, 5, 7}, d = {20, 27, 35} . x, y, z , xa, yb, zc d . 29. A, B, C D : ) A(2,1, 0); B(1,3,5); C (6, 4,3); D(0, 7,8) ) A(3,5, 1); B (7,5,3); C (9, 1,5); D(5,3, 3) . 30. AB : A(5,1,3); B (6, 2, 7) CD : C (4,1, 4); D(5, 0, 2) . M . 82

31. C (5, 0, 3); D(5, 2,1); E (1, 0,3) , AB : A(1, 0, 1); B (6, 2, 4) CDE . 32. AB : A(7, 4, 6); B (3, 2, 4) Oyz , Oxz , Oxy A1 , A2 , A3 . ) A2 A3 . ) A1 A2 : A2 A3 . 33. a, b, c | a |= 1, | b |= 2, | c |= 3 ,( a, b) = (b, c) = . p : 2 ( a, c ) = 4

) p = a + b + c ,

) p = a b + c .( a, b) = 2

34. a, b, c | a |= 2, | b |=| c |= 1 ,(c , b ) = q = a 3b . (a, c) = . 3

p = a+b+c

35. a b , q = 5a 4b .

p = a + 2b

36. a b . q = (ab)a | a |2 b p = xa + 3b x R a b .

37. a = m + n b = m n , | m |=| n |= 2 . 38. a = 2 p 3q b = p + q | p |= 2,| q |= 3 ( p, q ) = . 3

39. , a = 2 p + q b = p 2q | p |= | q |= 1 ( p, q ) = . 3

40. n p q | p |= 2 , | q |= 4 ,( p, q ) = , n p = 8 nq = 16 , : 3

) | n + q | ,

)

(n, p) .

41. c , a + b ab = 5, cb = 18, |b |= 2 . 42. a = {2, 2, 2} b = {0,1, 0} . x a , b . 3

83

43. Oxy x a = {1, 3, 2} .

10

44. Oz M A(3,5, 1) B (4,1, 7) . 45. A(5, 0, 0); B(0,5, 0) C (1, 2, 1) , M ' M (2,3, 4) ABC . 46. A(0, 3, 0); B (1, 3,1) C (0, 0, 3 ) , 2 M (3,5, 4) ABC . 47. M (2,3, 4) AB : A(1, 0, 1) ; B (4, 2, 0) . 48. p A(4, 2, 0) a = {1,3, 2} . M ' M (2,3, 4) p .

49. M ' M (4,1, 3) AB : A(5, 4, 6); B (3, 2, 2) . 50. | a |= 10, | b |= 5, ab = 12 , | [a, b] | . 51. | a |= 10, | b |= 5, |[a, b] |= 16 , ab . 52. [a, b] a = 2 p 3q + 5r , b = 4 p + q 2r p, q, r . 53. , AB = a + 2b AD = a 3b , | a |= 5, | b |= 3, (a, b) = . 3 54. | a |= 5 | b |,(a, b) = . 4

2b a 3a + 2b . 55. a b . x p = xa + 5b q = 3a b ? 56. 3m + 3n m n , | m |=| n |= 1 (m, n) = . 6 57. | [a, b] |2 +(ab)2 . 58. [a, b] = [c, d ] [a, c] = [b, d ] , a d b c . !

84

59. a, b, c . [a, b] + [b, c] + [c, a] . 60. [a, b] + [b, c] + [c, a] = o , a, b, c . ! 61. a = {1, 2,1}, b = {1, 1, 2}, c = {2,1, 1} . [a,[b, c]] b [a,[b, c]] . 62. a = {1,1, 1}, b = {2, 1, 2}, c = {1, 1, 2} . ) c a, b [ a, b ] .

) c a b . 63. x a = {2, 3,1} b = {1, 2,3} , c = {1, 2, 7} 10 3 6

.

64. ABC : ) A(2,1, 0); B(3, 6, 4); C (2, 4,1) ) A(1, 0,1); B(1,1, 0); C (0,1,1) . 65. a 1 , a 2 , a 3 . b 1 , b 2 , b 3 0, i j ai b j = 1, i = j.

66. : m = a +b+c , n = a +bc p = a b+c a, b, c (a, b, c) = 67. ABCD : A(2, 3,1); B (1,1,1); C (4,5, 6) D(2, 3, 6) . ? 68. ABCD : ) A(2, 1,1); B (5, 1, 2); C (3, 0, 3) D(6, 0, 1) , ) A(0, 0, 0); B (3, 4, 1); C (2,3,5) D(6, 0, 3) . 69. ABCD : A(2, 4,5); B (1, 3, 4); C (5,5, 1) D(1,1, 2) A . 70. [a,[b, c]] + [b,[c, a]] + [c,[a, b]] = o . 71. r a . 851 4

.

| [r ,[a, r ]] |=| [a, r ] | .

72. [[a, b],[c, d ]] = (a, c, d )b (b, c, d )a = (a, b, d )c (a, b, c)d . 73. : ) [[a, b],[c, d ]] [c,[[a, b], d ] , ) [a + b, b + c](a + c) . 74. [a, b][c, d ] = ac ad bc bd

: [a, b][c, d ] + c[[a, b], d ] .

86

III 1. 1.1. , S .

F ( x, y , z ) = 0

(1)

x, y, z . S (1) .. (1) S ( ), x, y z , S , x, y z , S . 1.2. . S , (1).

, (1) S , , S .1.3. .

( x a ) 2 + ( y b) 2 + ( z c ) 2 = r 2 C (a, b, c) r . , M ( x, y, z ) , MC = ( x a ) 2 + ( y b) 2 + ( z c) 2 .

(2)

, (2) , C r . , , (2), C r .

2. 2.1. . , x, y z .87

, F ( x, y, z ) = 0 G ( x, y, z ) = 0 , L , L , .. F ( x, y, z ) = 0 G ( x, y, z ) = 0 . ,

F ( x, y , z ) = 0 G ( x, y, z ) = 0 L .2.2. .

(3)

( x 1)2 + ( y 2)2 + ( z 3) 2 = 14 2 x + y2 + z2 = 1 ( ). 2.3. F ( x, y, z ) = 0 , G ( x, y, z ) = 0 H ( x, y, z ) = 0 ,

F ( x, y , z ) = 0 G ( x, y , z ) = 0 H ( x, y , z ) = 0 . , .2.4. . , (1, 1,0) 5, (1,1,3) 4, , Oxy Oz , 3 Oxy . .

( x + 1) 2 + ( y + 1)2 + z 2 = 25 2 2 2 ( x 1) + ( y 1) + ( z 3) = 16 z = 3 . z = 3

88

x 2 + y 2 + 2 x + 2 y = 14 2 x + y 2 2 x 2 y = 14. x + y = 0 , x 2 + y 2 = 14 , x = 7 , y = x . , ( 7, 7,3) ( 7, 7,3) .

3. 3.1. Oxyz . . S , Oz , : M 0 ( x0 , y0 , z0 ) S , , M 0 Oz , S .

S S .3.2. . Ox Oy .

Oz .3.3. .

F ( x, y ) = 0

(1)

, Oz .. M 0 ( x0 , y0 , z0 ) , S , (1). M 0 (1), ..

F ( x0 , y0 ) = 0 .

(2)

, M , M 0 Oz , S , .. , (1). M M 0 ( x0 , y0 , z0 ) Oz M 0 , , .. M ( x0 , y0 , z1 ) , z1 R . , (1) , 89

M (1). , S Oz . 3.4. . F ( x, y, z ) n , k R

F (kx, ky, kz ) = k n F ( x, y, z ) .

(3)

3.5. . S O , : M 0 ( x0 , y0 , z0 ) S O , M 0 O S . .

3.6. . F ( x, y, z ) n , F ( x, y , z ) = 0 (4) .

. M 0 ( x0 , y0 , z0 ) , O , (4).

F ( x0 , y0 , z0 ) = 0 .

(5)

M ( x, y, z ) OM 0 (4). OM OM 0 ( )

OM 0 o , k OM = kOM 0 , ..{x, y, z} = k{x0 , y0 , z0 }

x = kx0 , y = ky0 , z = kz0 . , x, y, z M

F ( x, y, z ) = F (kx0 , ky0 , kz0 ) = k n F ( x0 , y0 , z0 ) = k n i0 = 0 ,( F n M 0 ( x0 , y0 , z0 ) (5)). , S , (4), F ( x, y, z ) , .

3.7. . 90

F ( x, y , z ) = x 2 + y 2 z 2 ,

x2 + y 2 z 2 = 0 , .

4. 4.1. . x, y, z L

x = (t ), y = (t ), z = (t )

(1)

t , L .

4.2. . 4.1 L L . , , ( ) (1) . (1) t = 1 ( z )

x = ( 1 ( z )), y = ( 1 ( z )) L .

4.3. .

x = r cos t , y = r sin t , z = 0 ,

(2)

t [0, 2 ) . (2) Oxy . , (2)

x 2 + y 2 = r 2 cos 2 t + r 2 sin 2 t , z = 0.. x2 + y 2 = r 2 , z = 0. (3)

, (3) Oxy , r . 91

4.4. . x = ( p, q), y = ( p, q), z = ( p, q) . , , (4) p q . , , (4) p q x y : p = F ( x, y ) q = G ( x, y ) . (4) : z = ( F ( x, y ), G ( x, y )) = 0 (4) . (4)

5. 5.1. , . :Ax + By + Cz + D = 0 Ax + By + Cz + 2 Dxy + 2 Exz + 2 Fyz + 2Gx + 2 Hy + 2 Kz + L = 0 ..........................................................................................................2 2 2

(1) (2)

A, B, C , D, E , F ,... , . .

5.2. . S , . 5.3. . S , . 5.4. . S n , n . , .

92

5.5. . S n , S n . . , S F ( x, y , z ) = 0 (3) F ( x, y, z ) n , .. aklm x k y l z m k , l , m , k + l + m n , aklm , k , l , m n aklm . O ' x ' y ' z ' . , II 19.1 (7) II 19. S O ' x ' y ' z ' , (3) x, y, z (7) II 19.1. aklm (a + 11x '+ 21 y '+ 31z ')k (b + 12 x '+ 22 y '+ 32 z ')l (c + 13 x '+ 23 y '+ 33 z ')m , S O ' x ' y ' z ' n . Oxyz O ' x ' y ' z ' , , O ' x ' y ' z ' n , ( O ' x ' y ' z ' Oxyz , ). , S n .

5.6. . , .

6. 6.1. . . . . M 0 ( x0 , y0 , z0 ) n = { A, B, C} .93

M ( x, y, z ) . M 0 M n . , x, y, z . , M 0 M = {x x0 , y y0 , z z0 } n = { A, B, C} M 0 M n M 0 M n = 0 , .. A( x x0 ) + B ( y y0 ) + C ( z z0 ) = 0 . (1) (1) D = Ax0 By0 Cz0 Ax + By + Cz + D = 0 , (2) .

6.2. . . . (2), .. A, B, C . x0 , y0 , z0 (2). x0 , y0 , z0 (2), Ax0 + By0 + Cz0 + D = 0 . (2) (3) A( x x0 ) + B ( y y0 ) + C ( z z0 ) = 0 M 0 ( x0 , y0 , z0 ) n = { A, B, C} . (3)

6.3. n = { A, B, C} . A( x x0 ) + B ( y y0 ) + C ( z z0 ) = 0 M 0 ( x0 , y0 , z0 ) n = { A, B, C} . , Ax + By + Cz + D = 0 .

94

6.4. . , M 0 (1,1,1) n = {2, 2,3} . . 6.3 2( x 1) + 2( y 1) + 3( z 1) = 0 , .. 2 x + 2 y + 3 z 7 = 0 . 6.5. . M 0 r 0 n = { A, B, C},A2 + B 2 + C 2 0 . r M , M 0 M n , .. ( r r 0 ) n . , n( r r 0 ) = 0 , . (4)

6.6. , : A( x1, y1, z1 ); C ( x2 , y2 , z2 ); B( x3 , y3 , z3 ) , (. 1). M ( x, y , z ) , AB, AC AM . , AM {x x1, y y1, z z1}; AC{x2 x1, y2 y1, z2 z1}; AB{x3 x1, y3 y1, z3 z1} II 17.2 x x1 x2 x1 x3 x1 y y1 z z1 (5) y2 y1 z2 z1 = 0 y3 y1 z3 z1

A, B C . , A, B C n = [ AB, AC ] . , (5) n , A( x1, y1, z1 ) n .

6.7. . , A(1,1,0), B (2,0,2) C (0,1,1) . . (5) :

95

x 1 y 1 z 0 2 1 0 1 2 0 = 0 0 1 11 1 0 .. x + 3 y + z 4 = 0 .

6.8. M 0 ( x0 , y0 , z0 ) p = { p1, p2 , p3} q = {q1, q2 , q3} (. 2). , n = [ p, q] , M 0 M [ p, q] = 0 , .. ( M 0 M , p, q ) = 0 , M ( x, y , z ) . M 0 M = {x x0 , y y0 , z z0 } (6) M 0 p q x x0 p1 q1 y y0 p2 q2 z z0 p3 q3 (6)

= 0.

(7)

6.9. . M 0 ( x0 , y0 , z0 ) p = { p1, p2 , p3} q = {q1, q2 , q3} , M ( x, y , z ) M 0 M , p q . , M 0M = p + q .. {x x0 , y y0 , z z0 } = { p1, p2 , p3} + {q1, q2 , q3}

x = x0 + p1 + q1 y = y0 + p2 + q2 z = z + p + q , 0 3 3 .96

(8)

6.10. . A(1,1,1) p = {1,2,3} q = {3,2,1} . . . (7) :

x 1 y 1 z 1 1 3 .. 2 2 3 =0 1

x 2y + z = 0 . (8) : x = 1 + + 3 y = 1 + 2 + 2 z = 1 + 3 + .

7. . 7.1. 6.2

Ax + By + Cz = D ,

(1)

A2 + B 2 + C 2 0 . , .. A, B, C , D .7.2. . D = 0 , (1) Ax + By + Cz = 0 . . x = 0, y = 0, z = 0

Ax + By + Cz = 0 , . 7.3. . C = 0 , (1) Ax + By + D = 0 Oz Oz . . C = 0 . n Oz , .. n = { A, B, C} = { A, B,0} . , n k ,

97

Ax + By + D = 0 Oz Oz . 7.4. . B = C = 0 , (1) Ax + D = 0 Oyz . . B = C = 0 . n Oz Oy , .. n = { A, B, C} = { A,0,0} . ,

n k n j , Ax + D = 0 Oz Oy . , , , Ax + D = 0 Oyz . 7.5. . 7.3 7.4 :

) Ax + Cz + D = 0 Oy , By + Cz + D = 0 Ox , ) By + D = 0 Oxz , Cz + D = 0 Oxy .7.6.

Ax + By + Cz + D = 0

A2 + B 2 + C 2 + D 2 0 . Ax + By + Cz = D .

(2)

, (2) D 0 Ax D Cz + D + D = 1 By

..D A

x

+

yD B

+ zD = 1 .C

(3)

D a = D , b = D , c = C (3) A B

x a z + b + c =1. y

(4)

98

(4) . Ox y = z = 0 (4) x = a , Ox , a . , Oy Oz , , b c , . , (4) .7.7. . , a = 3, b = 2, c = 4 . . (4) :x + y 3 2 z + 4 =1

.. 4 x 6 y 3z + 12 = 0 .

8. . 8.1. 6.5 M 0 r 0 n = { A, B, C}, A2 + B 2 + C 2 0

n( r r 0 ) = 0 ,

(1)

r M . (1) n 0 = n , |n|

: n 0 (r r 0 ) = 0 . , n 0 ={ A, B ,C }A2 + B 2 + C 2

(2)

, r 0 = {x0 , y0 , z0 } r = {x, y , z} (2) Ax + By + Cz + D A2 + B 2 + C 2

, .. =0 (3)

D = Ax0 By0 Cz0 .99

8.2. . M1 ( x1, y1, z1 ) , (3), . 3. -

M 0 M 1 = r 1 r 0 , r 1 , r 0 - M 0 M1 . M 0 M1N1 , M1N 1 = h , h M1 ,

h =| prn ( r 1 r 0 ) |=0

|n 0 ( r 1 r 0 )| ||n 0 |

=| n 0 ( r 1 r 0 ) |=

| Ax1 + By1 + Cz1 + D | A2 + B 2 + C 2

.

(4)

, M1 ( x1, y1, z1 ) , (3), M1 (3) . O (0,0,0) (3) ho =|D | A + B2 +C 22

.

8.3. . M (2,4,1) :

4 x 5 y + z 12 = 0 A = 4, B = 5, C = 1 , (5)

(5)

. (5) .

A2 + B 2 + C 2 = 42 + ( 5)2 + 12 = 42 . ,4 x 5 y + z 12 42

=0.

, (4), M : h =|4i 2 5i 4 +1i112| 42

= 23 . 42

9. 9.1. '

Ax + By + Cz + D = 0 , A ' x + B ' y + C ' z + D ' = 0 , (1) . ' n = { A, B, C} n ' = { A ', B ', C '} n n ' 100

' , . 4. , ' cos = nn ' ,|n|i|n '|

(2)

nn ' = AA '+ BB '+ CC ' , | n |=

A2 + B 2 + C 2 | n ' |=

A '2 + B '2 + C '2 .

9.2. . , ' cos = 0 , ..

AA '+ BB '+ CC ' = 0 .

(3)

9.3. . ' cos = 1 , .. n n ' , A' A

= B' = C' . B C

(4)

, (1) (4), , .. . .9.4. .

x z = 0, y z = 0.. n = {1,0, 1} n ' = {0,1, 1} ,

cos = nn ' =|n|i|n '|

1i 0 + 0i1+ ( 1)i ( 1) 1 + 02 + ( 1)2 02 +12 + ( 1)22

=1, 2

.. = . 3

10. 10.1. . L M 0 ( x0 , y0 , z0 )

s = {l , m, n} . M ( x, y , z ) L , M 0 M s , .. M 0 M = ts , t R . , M 0 M = r r 0 , r r 0 - M M 0 , , . 5, r r 0 = ts, t R

101

.. r = r 0 + ts, t R (1)

L M 0 s .

10.2. . - r r 0 M M 0 L r 0 = {x0 , y0 , z0 } r = {x, y , z} (1) {x, y , z} = {x0 , y0 , z0 } + t{l , m, n} , t R , {x, y , z} = {x0 + tl , y0 + tm, z0 + tn} , t R . x = x0 + tl y = y0 + tm z = z + tn, t R 0 L . (2)

10.3. . (4) t L . , x x0 l x x0 l

= t,

y y0 m zz

= t,

z z0 n

= t , t R , .. (3)

= m0 = n0 .

y y

10.4. . l , m, n s , (3) . , n 0 , x x0 = z z0 l n (4) y y zz 0 = n0 m L . (4) y , x . , Oy , Ox , .. L Oxz Oyz , .

10.5. . , , L , .. s = {l , m, n} . 102

l =| s | cos , m =| s | cos , n =| s | cos . l , m, n (3) x x0 |s|cos

=

y y0 |s |cos

=

z z0 |s |cos

| s | L , .. x x0 cos 0 0 = cos = cos .

y y

zz

(5)

10.6. . M (3, 4,5) Oz . . s = k = {0,0,1} (3) x 3 0

= 0 = z 5 . 1

y +4

(6)

(6) cos = 0, cos = 0, cos = 1 , Ox Oy . , (6) x 3 = 0 y + 4 = 0

x 3 = 0 y + 4 = 0. 10.7. . L M 0 ( x0 , y0 , z0 ) M1 ( x1, y1, z1 ) , M 0 M1 L (. 6).

, L M 0 M 1 = {x1 x0 , y1 y0 , z1 z0 } M 0 ( x0 , y0 , z0 ) . , s = {l , m, n} , l = x1 x0 , m = y1 y0 , n = z1 z0 (3) x x0 x1 x0

= y y0 = z z0 . 1 0 1 0

y y

zz

(7)

10.8. . L M 0 (1,1,1) M1 (2,3,4) . (7)

103

x 1 2 1

y 1 z y 1 = 31 = 4 1 , .. x1 1 = 2 = z 3 1 . 1

10.9. , ' , (. 7). L = '

Ax + By + Cz + D = 0

A' x + B ' y + C ' z + D ' = 0

' . n = { A, B, C} n ' = { A ', B ', C '} ' , , L . , s , L s n s n ' , s || [n, n '] , s = [n, n '] . , L L , ..

Ax + By + Cz + D = 0 A' x + B ' y + C ' z + D ' = 0 (3).

(8)

10.10.. L :

x 2 y + 3z 4 = 0, 3x 2 y + z = 0.. n = {1, 2,3} n ' = {3, 2,1} . ,i j k s = [n, n '] = 1 2 3 = {4,8,4} . 3 2 1 (9) 2x 2z + 4 = 0 .

(9)

(10)

(10) x = 0 z = 2 . (10) 0 2 y + 6 4 = 0 , .. y = 1 . , L M (0,1,2) . ,

104

x 0 4

= 8 = z 2 . 4

y 1

(11)

10.11. . , (8) (7). , (10) z = 0 2 x + 4 = 0 , .. x = 2 . (9) y = 3 . , L M 0 (0,1,2) M1 ( 2, 3,0) , x 0 2 0 y 1 y 1 z = 31 = 0 2 , .. x20 = 4 = z22 . 2

, (11), s = {4,8,4} s ' = {2, 4, 2} .

11. 11.1. L L1 x x0 l x x1 l1

= m0 = n0 = m1= n1 1 1y y zz

y y

zz

(1) (2)

. L M 0 ( x0 , y0 , z0 ) s = {l , m, n} , L1 M1 ( x1, y1, z1 ) s 1 = {l1, m1, n1} . M 0 M 1 , s s 1 . : ) M 0 M 1 , s s 1 , .. ( M 0 M 1, s, s 1 ) = 0 L L1 , ) M 0 M 1 , s s 1 , .. ( M 0 M 1, s, s 1 ) 0 L L1 , .. . .

11.2. M 0 M 1 , s s 1 , .. ( M 0 M 1, s, s 1 ) = 0 . :

105

1) s s 1 , .. s 1 = k s 2) s s 1 . , L L1 , , M 0 ( x0 , y0 , z0 ) s s 1 , 6.8 x x0 l l1 y y0 m m1 z z0 n n1 =0.

, L L1 ( M 0 M , M 0 M 1 , s ) = 0 , .. x x0 x1 x0 l y y0 y1 y0 m z z0 z1 z0 = 0 . n (3)

1) , M 0 L1 , .. (2), L L1 . , M 0 L1 , L L1 (. 8). , L L1 M 0 L1 , .. H M 0 M 1 s 1 . , H=|[ M 0 M 1 , s1 ]| |s 1 |

.

(4)

2) L L1 . , L L1 s s 1 (. 9) cos =ss1|s|i|s1 |

.

(5)

106

, A L L1 : -

x = x0 + l x = x1 + l1 y = y0 + m y = y1 + m1 z = z + n z = z + n 1 1 0 , , - x = x y = y

x0 + l = x1 + l1 y0 + m = y1 + m1 , - L ( L1 ) A . 11.3. . L L1 x 1 = y +1 = z 2 1 3 2 y +1 x + 2 = 6 = z4 1 , 2

. L L1 . . L M 0 (1, 1, 2) s = {1,3, 2} ,

L1 M1 (2, 1,1) s 1 = {2,6, 4} . , M1M 0 = {3,0,1} ,

i

j k

[ M1M 0 , s 1 ] = 3 0 1 = {6, 10,18} . 2 6 4 , (3) : H=|[ M 0 M 1 , s1 ]| |s 1 |

=

( 6)2 + ( 10)2 +182 2 +6 + 42 2 2

= 460 = 115 . 56 14

11.4. . x 1 a 5 = 1 = z23 y +1

(L) ( L')107

x 1 = y +1 = z 1 . a +1 10 4

) a L L ' . ) a L L ' . ) L L ' . ) .. ) L A(1 + a, 1,3) s = {5,1, 2} , L '

B (1, 1,1) s ' = {10,1 + a, 4} , AB = { a,0, 2} . L L ' ( AB, s, s ') = 0 , .. a

0 1 a +1

2 2 = 0 , 4

5 10

a 2 + 4a 5 = 0 . a1 = 5 a2 = 1 . a2 = 1 s ' = {10, 2, 4} = 2i{5,1, 2} = 2s , . a1 = 5 s ' = {10, 4, 4} , :x+4 5 = 1 = z23 y +1

(L) ( L')

x 1 = y +1 = z 1 . 10 4 4

) L : x = 5t 4 y = t 1 z = 2t + 3 L ' : x = 10 + 1 y = 4 1 z = 4 + 1. , x = x y = y :

5t 4 = 10 + 1 t 1 = 4 1.

108

= 1 , t = 2 6 34 2 L A ' x = 6 , y = 6

z = 10 , .. A '( 2 , 1 , 5 ) . 3 3 3 6 ) L A(4, 1,3) s = {5,1, 2} , L ' B (1, 1,1) s ' = {10, 4, 4} , L L ' : x+4 5 10 .. 2 x + 5 z 7 = 0 . y +1 1 4 z 3 2 = 0 4

) 2 x + 5 z 7 = 0 y , Oy . a = 1 L L ' :x 5 = 1 = z23 x 1 = 2 = z 1 , 10 4 y +1 y +1

. L A(0, 1,3) s = {5,1, 2} , L ' B (1, 1,1) s ' = {10, 2, 4} . , x 1 5 .. 2 x 12 y + z 15 = 0 . : M Ox : y = z = 0 x = 15 , .. 2 M (15 ,0,0) , 2 15 N Oy : x = z = 0 y = 12 , .. 15 N (0, 12 ,0) ,

y +1 0 1

z 3 2 = 0 , 2

-

P Oz : x = y = 0 z = 15 , .. P (0,0,15) .

15 OM = {15 ,0,0} , ON = {0, 12 ,0} OP = {0,0,15} , 2

109

15 2

015 12

0 0 = 375 . 16 15

V = 1 | (OM , ON , OP ) |= 1 0 6 6 0

0

11.5. M 0 M 1 , s, s 1 ,

L L1 . L L1 L * L L1 L L1 . L * L L1 . .11.6. . x 1 x 1 = 3 = z23 y 3 y

(L) ( L')

= 1 = z1 . 1

L * L L ' .. L * x x0 l

= m0 = n0 .

y y

zz

(5)

L : A(0,3,3) s = {1, 3, 2} , L ' : B (0,0,1) s ' = {1,1,1} . L L * L ' L * s '' = {l , m, n} i [ s, s '] = 1 1 j 3 1 k 2 = {1, 3, 4} . 1

L L * , AM , s, s '' , .. ( AM , s, s '') = 0 , x0 1 1 .. 9 x0 + y0 + 3z0 12 = 0 . (6) y0 3 3 3 z0 3 2 = 0 4

110

L ' L * , BM , s ', s '' , .. ( BM , s ', s '') = 0 , x0 1 1 .. 7 x0 + 5 y0 + 2 z0 2 = 0 . x0 = 0 (6) (7) y0 + 3 z0 12 = 0 5 y0 + 2 z0 2 = 058 58 : y0 = 18 , z0 = 13 . , M ( x0 , y0 , z0 ) M (0, 18 , 13 ) 13 13 (5) 18 x = y + 13 1 3

y0 3 1

z0 1 4 =0 1 (7)

= 413 .

z 58

12. 12.1. L :x x0 l

= m0 = n0 .

y y

zz

(1) (2)

Ax + By + Cz + D = 0 .

M 0 ( x0 , y0 , z0 ) s = {l , m, n} , n = { A, B, C} . ns . : ) ns = 0 , L M 0 . , M 0 , L L M 0 , .. H =| Ax0 + By0 +Cz0 + D| A2 + B 2 +C 2

, (. 10).

111

) ns 0 , L , L , (. 11). : x x0 = y y0 = z z0 l m n Ax + By + Cz + D = 0.

(3)

(3) (?).12.2. . L , L ' n ' M 0 ' , .. M 0 M n = 0 , M ( x, y, z ) ' .

, L , L | | , s n , 2 .. cos = ns .|n|i|s|

12.3. . L

+ = z1 1 2x + y z + 4 = 0 ,x 1 = y 2 1 1

(4) (5)

. L . :y 2 + x1 1 = 1 = z1 1 = t 2 x + y z + 4 = 0

.. x = t + 1, y = t + 2, z = t 1 2 x + y z + 4 = 0 2(t + 1) + (t + 2) (t 1) + 4 = 0 2t + 9 = 0 t = 9. 2112

, x = 9 + 1 = 7 , y = 9 + 2 = 5 , z = 9 1 = 11 , .. 2 2 2 2 2 2 A( 7 , 5 , 11) . 2 2 2 L , s = {1,1,1} n = {2,1, 1} cos = ns = 2+11 = 32 , .. = arccos 32 , |n|i|s| 3 6

= arccos 2 . 2 2 3

13. 13.1. . L x4 1 z = 3 =2 y 2

(1)

M (2,3, 4) . M1 M L .. L M 0 (4, 2,0) s = {1,3, 2} . M1 M L , M * MM1 MM * s , . 12. , L

x = t + 4 y = 3t + 2 z = 2t , t R M * (t + 4,3t + 2, 2t ) , t R . , MM * = {t + 2,3t 1, 2t 4} MM * s MM * s = 0 , .. t + 2 + 3(3t 1) + 2(2t 4) = 0 14t 9 = 09 t = 14 . 9 65 9 55 9 18 65 55 18 , x = 14 + 4 = 14 , y = 3i14 + 2 = 14 , z = 2i14 = 14 , .. M * ( 14 , 14 , 14 ) . -

, M1 ( x, y, z ) , MM * = M * M 1 , 65 55 18 65 55 18 {14 2, 14 3, 14 4} = {x 14 , y 14 , z 14}

113

.. x = 51 , y = 34 , z = 10 M L 7 7 7 M1 ( 51 , 34 , 10 ) . 7 7 7

13.2. . S (3, 1, 2) L L1 x +5 2 x 3 2 + = 4 = z31 y 3

(2) (3)

= 3 = z 2 , 4

y +1

. S L L1 .

. S L . L1 , .. A = L1 . L * S A (. 13). L M (5,3, 1) s = {2, 4,3} . X ( x, y , z ) , SX = {x + 3, y + 1, z 2}, SM = {2, 4, 3} s , ( SX , SM , s ) = 0 , .. x+3 2 2 y +1 4 4 z2 3 = 0 . 3 (4)

, X . , (4) , .. SM s . S L (!). , , A L1 SA . , A(3, 1, 2) x 3 33 y +1 y +1 z = 1+1 = 22 , .. x63 = 0 = z 2 . 0 2

114

13.3. . L x = 5t + 3 y = t 1 z = t + 4, t R : 2 x 2 y + 3z + 5 = 0 . (6) (5)

. L x 3 5

= 1 = z 4 , 1

y +1

(7)

L M (3, 1, 4) s = {5,1,1} . , n = {2, 2,3} . L L ' 1 L

, . 15. , 1 M s n , 6.8 x3 5 2 .. 5 x 13 y 12 z + 20 = 0 . , L 2 x 2 y + 3z + 5 = 0 5 x 13 y 12 z + 20 = 0. y +1 1 2 z4 1 3 =0

13.4. . Q (3,0, 2) L 3x + 2 y + 3z 5 = 0 x + y + z 4 = 0. (8)

. s L s = [n, n '] , n = {3, 2,3} n ' = {1,1,1} L . ,

115

i s= 3 1

j 2 1

k 3 = {1,0,1} . 1

M L (8) x = 0 2 y + 3z 5 = 0 y + z 4 = 0. 2 z + 3 = 0 , .. z = 3 . y = 7 M M (0,7, 3) . X ( x, y , z ) (. 16), MX , MQ s , .. ( MX , MQ, s ) = 0 x 3 1 y7 7 0 z +3 5 1 = 0 , .. 7 x + 2 y + 7 z + 7 = 0 .

13.5. . x = 5 + 4t y = 2 + 7t z = 1 + 2t , t R L1 x = 3 + 2t y = 4 + t z = 2 3t , t R. (10) (9)

. (10) L1 M (3, 4, 2) s ' = {2,1, 3} , (9) L N (5, 2,1) s = {4,7, 2} . , M s s ' , 6.8

116

x3 2 4

y+4 1 7

z2 3 = 0 , .. 23 x 16 y + 10 z 153 = 0 . 2

13.6. . L

= z +3 5 x + y + 4z 9 = 0

x 1 = y 2 4 2

(11) (12)

. L1 L . . L M (1, 2, 3) s = {2, 4,5} , n = {1,1, 4} . sn = 18 0 , . 18.

M L * , .. n . x 1 = y 2 1 1

= z +3 = t . 4

(13)

M ' = L * . (13) x = t + 1, y = t + 2, z = 4t 3 (12)

t + 1 + t + 2 + 4(4t 3) 9 = 0 , .. t = 1 . , M ' x = 2, y = 3, z = 1 , .. M '(2,3,1) . MM ' = M ' M 1 M1 , M . , {1,1, 4} = {x 2, y 3, z 1} , .. x = 3, y = 4, z = 5 M1 (3, 4,5) . N = L . (11) x = 2t + 1, y = 4t + 2, z = 5t 3 (12) 2t + 1 4t + 2 + 4(5t 3) 9 = 0 , .. t = 1 . , N x = 2i1 + 1 = 3, y = 4i1 + 2 = 2, z = 5i1 3 = 2 , .. N (3, 2, 2) . , L * L 117

x 3 33

y + 2 z 2 y+2 = 4+ 2 = 5 2 , .. x0 3 = 6 = z 2 . 3

L x = 2 + 3t y = 4 2t z = 8 + 2t.

13.7. .

3x 2 y 3z 7 = 0

(14) (15)

L * M (3, 2, 4) , L .. L * , 1 . , 1

n = {3, 2, 3} M 1 nMX = 0 , .. {x 3, y + 2, z + 4}i{3, 2, 3} = 0 , 1 3x 2 y 3z 25 = 0 . N = L 1 . 3(2 + 3t ) 2(4 2t ) 3(8 + 2t ) 25 = 0 7t 51 = 0 t = 51 . 7 , N :51 51 51 x = 2 + 3i7 = 167 , y = 4 2i7 = 74 , z = 8 + 2i7 = 158 , 7 7 7

(16)

N (167 , 74 , 158 ) . , L * M N 7 7 7 167 3 7

x 3

=

y+2 74 + 2 7

y+2 z+ + = 158 4 , .. x733 = 30 = z934 . +47

14. 14.1. . , a11x2 + a22 y2 + a33 z 2 + 2a12 xy + 2a23 yz + 2a13 xz + 2a14 x + 2a24 y + 2a34 z + a44 = 0 (1)118

a11, a22 , a33 , a12 , a23 , a13 . , (1) .14.2. . , (1) , , . 14.3. . (1) a11, a22 , a33 , a12 , a23 , a13 (1), a14 , a24 , a34 (1), a44 (1).

x = x '+ x0 y = y '+ y0 z = z '+ z 0 (2)

x0 , y0 , z0 O ' (?). (2) (1)

a11 x '2 + a22 y '2 + a33 z '2 + 2a12 x ' y '+ 2a23 y ' z '+ 2a13 x ' z '' ' ' ' + 2a14 x '+ 2a24 y '+ 2a34 z '+ a44 = 0

(3)

' a14 = a11 x0 + a12 y0 + a13 z0 + a14 ' a24 = a12 x0 + a22 y0 + a23 z0 + a24 ' a34 = a13 x0 + a23 y0 + a33 z0 + a34 ' 2 2 2 a44 = a11x0 + a22 y0 + a33 z0 + 2a12 x0 y0 + 2a23 y0 z0 + 2a13 x0 z0 + 2a14 x0 + 2a24 y0 + 2a34 z0 + a44 .

(4)

.. (1) , (1) (4).

119

14.4. . , :

x = m11x '+ m12 y '+ m13 z ' y = m21x '+ m22 y '+ m23 z ' x = m x '+ m y '+ m z ' 31 32 33

(5)

mij = m ji , i = 1, 2,3; j = 1, 2,3 . (5) (1) x ', y ', z ' S O ' x ' y ' z ' . ' ' ' ' ' ' a11 x '2 + a22 y '2 + a33 z '2 + 2a12 x ' y '+ 2a23 y ' z '+ 2a13 x ' z ' ' ' ' + 2a14 x '+ 2a24 y '+ 2a34 z '+ a44 = 0 ' , aij .

(6)

. (6) mij , (5), (1);

(6) mij (1), . , (1) , (6) . 14.5. . 14.3 (1), , (1). 14.6. . O ' x ' y ' z ' ( Oxyz ), (3) ' ' ' S 2a14 x ', 2a24 y ', 2a34 z ' , .. ' ' ' a14 , a24 , a34 . (4) x0 , y0 , z0

120

a11x0 + a12 y0 + a13 z0 + a14 = 0 a12 x0 + a22 y0 + a23 z0 + a24 = 0 a13 x0 + a23 y0 + a33 z0 + a34 = 0.

(7)

(7) S , O ' ( x0 , y0 , z0 ) , x0 , y0 , z0 (7) S . , S O ' , .. (7) ( x0 , y0 , z0 ) . O ' . S O ' x ' y ' z ' : a11 x '2 + a22 y '2 + a33 z '2 + 2a12 x ' y '+ 2a23 y ' z '+ 2a13 x ' z '+ a44 = 0 . , M ( x ', y ', z ') S , M * ( x ', y ', z ') , M O ' S . , S O ' , O ' . , (7). , , .14.7. . S

2a12 x ' y ', 2a23 y ' z ', 2a13 x ' z ' .. F

(1), .. F ( x , y , z ) = a11 x 2 + a22 y 2 + a33 z 2 + 2 a12 xy + 2 a23 yz + 2 a13 xz (8) F x, y, z x2 + y 2 + z 2 = 1 , .. 1. P F ( x, y, z ) ( P F ( x, y, z ) ). Oz ' -

121

O P , Ox ' Oy ' Oz ' . , Ox ' y ' z ' P (0,0,1) . Ox ' y ' z ' F ' ' ' ' ' ' F = a11 x '2 + a22 y '2 + a33 z '2 + 2a12 x ' y '+ 2a23 y ' z '+ 2a13 x ' z '

(10)

x '2 + y '2 + z '2 = 1 , (11) F (10) ( x ', y ', z ') (11). F (0,0,1) .' ' , (10) a23 a13 ' . , , a13 = 0 . F K (11) y ' = 0 , .. Ox ' z ' . - M K Oz ' . M

x ' = sin , y ' = 0, z ' = cos .

(12)

(12) (10) F K:' ' ' F = a11 sin 2 + a33 cos2 + 2a13 sin cos =' ' a11 +a33 2

+

' ' a33 a11 cos 2 2

' + a13 sin 2

(13)

, F K (13) . = 0 , (12), = 0 (0,0,1) , F . (13) = 0 . (13) , ' ' ' ' F ' = ( a33 a11 )sin 2 + 2a13 cos 2 = 0 2a13 = 0 , ' ' .. a13 = 0 . , a23 = 0 F K * , (11) x ' = 0 . , Ox ' y ' z ' F S : ' ' ' ' F = a11 x '2 + 2a12 x ' y '+ a22 y '2 + a33 z '2 ,

(14)

Ox ' Oy ' , Ox ', Oy ' Oz ' 122

. , Ox ' y ' z ' Oz ' (14). x ' y ' , z ' . ' , a12 x ' y ' .

14.8. . Ox ' y ' z ' (1) S :' ' ' ' ' ' ' a11 x '2 + a22 y '2 + a33 z '2 + 2a14 x '+ 2a24 y '+ 2a34 z '+ a44 = 0 .

(15)

14.9. . (1) S (15) . 14.10. . 14.7 x ' y ' . x '' y '' ,

x ' = x ''cos y ''sin y ' = x ''sin + y ''cos

(16)

' a12 (14) ,

ctg 2 =

' ' a11 a22 ' 2 a12

x '' y '' , .

15. 15.1. S S . 14.8 S ' ' ' ' ' ' ' a11 x '2 + a22 y '2 + a33 z '2 + 2a14 x '+ 2a24 y '+ 2a34 z '+ a44 = 0

, S , 123

' ' a11 x '+ a14 = 0 ' ' a22 y '+ a24 = 0 ' ' a33 z '+ 2a34 = 0

(1)

S ' ' ' '' a11 x ''2 + a22 y ''2 + a33 z ''2 + a44 = 0

, 14.6 (1) , ' aii 0 , i = 1, 2,3 . , S ,

a11 x 2 + a22 y 2 + a33 z 2 + a44 = 0

(2)

aii 0 , i = 1, 2,3 ( Oxyz ). , aii , i = 1, 2,3, 4 .15.2. aii , i = 1, 2,3 (2) , a44 , S .

aii , i = 1, 2,3, 4 , (2) x, y, z R , .. S . S . aii , i = 1, 2,3 a44 , S . . , , , a44 , a44 , a44 , 11 22 33 a 2 , b 2 , c 2 , . x2 a2a a a

+

y2 b2

2 + z2 = 1

c

(3)

. , (3), 124

Ox, Oy, Oz .15.3. aii , i = 1, 2,3, 4 , , S .

. a11 > 0, a22 > 0, a33 < 0, a44 < 0 . a44 , 11 a44 , a44 , a 2 , b 2 , c 2 . 22 33

a

a

a

(2) :x2 a2

+

y2 b2

2 z2 = 1

c

(4)

. , (4), Ox, Oy, Oz .15.4. a11 , a22 , a33 a11 , a22 , a33 , a44 , S .

. a11 < 0, a22 < 0, a33 > 0, a44 < 0 . a44 > 0, a44 > 0, a44 > 0 11 22 33 a 2 , b 2 , c 2 . (2) :x2 a2a a a

+

y2 b2

2 z 2 = 1

c

(5)

. , , Ox, Oy, Oz .15.5. a44 , S .

aii , i = 1, 2,3 , (2) x = y = z = 0 , .. S , . S .

125

aii , i = 1, 2,3 , S . a11 > 0, a22 > 0, a33 < 0 a1 , a1 , a1 a 2 , b 2 , c 2 11 22 33 . (2) :x2 a2

+

y2 b2

2 z2 = 0

c

(6)

.

16. 16.1. S , .. (7) 14.6 , a11 I 3 = a12 a13 a12 a22 a23 a13 a23 = 0 . a33

(1)

S , ' ' ' ' ' ' ' a11 x '2 + a22 y '2 + a33 z '2 + 2a14 x '+ 2a24 y '+ 2a34 z '+ a44 = 0 .

(2)

I 3 = 0 I 3 (2) ' ' ' ' ' ' a11a22 a33 = 0 , a11 , a22 , a33 . ' ' ' a11 , a22 , a33 . ' ' ' 16.2. a11 , a22 , a33 , ' a33 = 0 . x ', y ' z ' x, y, z ,

x = x '+

' a14

' a11

, y = y '+

' a24 ' a22

, z = z'.

(3)

(3) x ', y ' z ' , (2) ' ' ' ' a11 = a11 , a22 = a22 , a34 = p, a44 = q , S Oxyz

126

a11x 2 + a22 y 2 + 2 pz + q = 0 .) p = q = 0 . , S

(4)

x a22 y = 0 , 11 , a11 a22 , a11 a22 . ) p = 0, q 0 . (4) a11x 2 + a22 y 2 + q = 0 . (5)

a

, 3.3, (5) Oz . , a11 , a22 , q , (5) x y , .. . a11 , a22 , q , . , , a11 a22 , q , a 11q a . a 2 b 2 , , 22 (5) q

x2 a2

+

y2 b2

=1.

(6)

, . a11 a22 , , x2 a2

y2 b2q

=1.

(7)

) p 0 . O '(0,0, 2 p ) . x, y, z . , S , (4) z z 2 p , a11x 2 + a22 y 2 + 2 pz = 0 , (8)q

. , a11 a22 ,

127

2 y2 z = x2 + 2 .

a

b

(9)

a11 a22 , 2 y2 z = x2