Задача 3. Найти косинус угла между векторами AB и AC.

3.1. A(1,

−2, 3),

B(0,

−1,

2)

,

C (3, −4,

5).

3.2. A(0,

−3, 6),

B(−12,

−3,

−3), C (−9,

−3, −6).

3.3. A(3,

3,

−1),

B(5,

5,

−2),

C (4,

1,

1).

3.4. A(−1,

2,

−3),

B(3,

4, −6),

C (1,

1,

−1).

3.5. A(−4,

−2, 0),

B(−1,

−2,

4), C (3,

−2,

1).

3.6. A(5,

3,

−1),

B(5,

2,

0),

C (6,

4,

−1).

3.7. A(−3,

−7, −5),

B(0,

−1,

−2), C (2,

3,

0).

3.8. A(2,

−4, 6),

B(0,

−2, 4),

C (6,

−8,

10).

3.9. A(0,

1,

−2),

B(3,

1,

2),

C (4,

1,

1).

3.10. A(3,

3, −1),

B(1,

5,

−2), C (4,

1,

1).

3.11. A(2,

1,

−1),

B(6,

−1, −4), C (4, 2,

1).

3.12. A(−1,

−2, 1),

B(−4,

−2,

5), C (−8,

−2, 2).

3.13. A(6,

2, −3),

B(6,

3,

−2),

C (7,

3,

−3).

3.14. A(0,

0,

4),

B(−3,

−6,

1),

C (−5,

−10,

−1).

3.15. A(2,

−8, −1)

,

B(4,

−6,

0), C (−2,

−5, −1).

3.16. A(3,

−6, 9),

B(0,

−3, 6),

C (9,

−12,

15).

3.17. A(0,

2, −4),

B(8,

2,

2),

C (6,

2,

4).

3.18. A(3,

3, −1),

B(5,

1,

−2), C (4,

1,

1).

3.19. A(−4,

3,

0),

B(0,

1,

3), C (−2,

4,

−2).

3.20. A(1,

−1, 0),

B(−2, −1,

4), C (8,

−1,

−1).

3.21. A(7,

0,

2),

B(7,

1,

3),

C (8, −1,

2).

3.22. A(2,

3,

2),

B(−1,

−3, −1),

C (−3,

−7,

−3).

3.23. A(2,

2,

7),

B(0,

0,

6),

C

(−2,

5,

7).

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3.24. A(−1,	2,	−3), B(0,			1,	−2), C (−3,				4, −5).
3.25. A(0,	3,	−6),	B(9,	3,		6), C (12,			3,		3).
3.26. A(3,	3,	−1),	B(5,	1,		−2), C (4,			1, −3).
3.27. A(−2,	1,	1),	B(2,	3,		−2), C (0,			0,	3).
3.28. A(1,	4,	−1),	B(−2, 4,				−5), C (8,		4,		0).
3.29. A(0,	1,	0),	B(0,	2,	1),		C (1,	2,	0).
3.30. A(−4,	0,	4),	B(−1, 6,				7), C (1,		10,		9).
3.31. A(−2,	4,	−6), B(0,			2,		−4),	C (6,		8,	−10).

Задача 4. Вычислить площадь параллелограмма, построенного на векторах a и b.

4.1. a = p + 2q,

b = p + 2q;

p

=1,

q

= 2,

(p q) = π 6.

4.2. a = 3p + q,

b = p − 2q;

p

= 4,

q

=1,

(p q) = π 4.

4.3. a = p − 3q,

b = p + 2q;

p

=1 5,

q

=1,

(p q) = π 2.

4.4. a = 3p − 2q,

b = p + 5q;

p

= 4,

q

=1 2,

(p q) = 5π 6.

4.5. a = p − 2q,

b = 2p + q;

p

= 2,

q

= 3,

(p q) = 3π 4.

4.6. a = p + 3q,

b = p − 2q;

p

= 2,

q

= 3,

(p q) = π 3.

4.7. a = 2p − q,

b = p + 3q;

p

= 3,

q

= 2,

(p q) = π 2.

4.8. a = 4p + q,

b = p − q;

p

= 7,

q

= 2, (p q) = π 4.

4.9. a = p − 4q,

b = 3p + q;

p

=1,

q

= 2,

(p q) = π 6.

4.10. a = p + 4q,

b = 2p − q;

p

= 7,

q

= 2,

(p q) = π 3.

4.11. a = 3p + 2q, b = p − q;

p

=10,

q

=1,

(p q) = π 2.

4.12. a = 4p − q,

b = p + 2q;

p

= 5,

q

= 4,

(p q) = π 4.

227

4.13. a = 2p + 3q,

b = p − 2q;

p

= 6,

q

= 7,

(p q) = π 3.

4.14. a = 3p − q,

b = p + 2q;

p

= 3,

q

= 4,

(p q) = π 3.

4.15. a = 2p + 3q,

b = p − 2q;

p

= 2,

q

= 3,

(p q) = π 4.

4.16. a = 2p − 3q,

b = 3p + q;

p

= 4,

q

=1,

(p q) = π 6.

4.17. a = 5p + q,

b = p − 3q;

p

=1,

q

= 2,

(p q) = π 3.

4.18. a = 7p − 2q,

b = p + 3q;

p

=1 2,

q

= 2,

(p q) = π 2.

4.19. a = 6p − q,

b = p + q;

p

= 3,

q

= 4, (p q) = π 4.

4.20. a =10p + q,

b = 3p − 2q;

p

= 4,

q

=1,

(p q) = π 6.

4.21. a = 6p − q,

b = p + 2q;

p

= 8,

q

=1 2,

(p q) = π 3.

4.22. a = 3p + 4q,

b = q − p;

p

= 2,5,

q

= 2,

(p q) = π 2.

4.23. a = 7p + q,

b = p − 3q;

p

= 3,

q

=1,

(p q) = 3π 4.

4.24. a = p + 3q,

b = 3p − q;

p

= 3,

q

= 5,

(p q) = 2π 3.

4.25. a = 3p + q,

b = p − 3q;

p

= 7,

q

= 2,

(p q) = π 4.

4.26. a = 5p − q,

b = p + q;

p

= 5,

q

= 3, (p q) = 5π 6.

4.27. a = 3p − 4q,

b = p + 3q;

p

= 2,

q

= 3,

(p q) = π 4.

4.28. a = 6p − q,

b = 5q + p;

p

= 1 2,

q

= 4,

(p q) = 5π 6.

4.29. a = 2p + 3q,

b = p − 2q;

p

= 2,

q

=1,

(p q) = π 3.

4.30. a = 2p − 3q,

b = 5p + q;

p

= 2,

q

= 3,

(p q) = π 2.

4.31. a = 3p + 2q,

b = 2p − q;

p

= 4,

q

= 3,

(p q) = 3π 4.

Задача 5. Компланарны ли векторы a, b и c?

5.1. a = {2, 3, 1}, b = {−1,

0,

−1},

c = {2,

2, 2}.

228

5.2. a = {3,

2,

1} ,

b = {−1,

1,

−1} ,

c = {3,

1,

−1}.

5.3. a = {1,

5,

2} ,

b = {−1,

1,

−1} ,

c = {1,

1,

1}.

5.4. a = {1,

−1,

−3}

,

b = {3,

2,

1} ,

c = {2,

3,

4}.

5.5. a = {3,

3,

1} ,

b = {1,

−2,

1} ,

c = {1,

1,

1}.

5.6. a = {3,

1,

−1} ,

b = {−2,

−1, 0} ,

c = {5,

2,

−1}.

5.7. a = {4,

3,

1} ,

b = {1,

−2,

1} ,

c = {2,

2,

2}.

5.8. a = {4,

3,

1} ,

b = {6,

7,

4} ,

c = {2,

0,

−1}.

5.9. a = {3,

2,

1} ,

b = {1,

−3,

−7} ,

c = {1,

2,

3}.

5.10. a = {3,

7,

2} ,

b = {−2,

0, −1} ,

c = {

2,

2,

1}.

5.11. a = {1,

−2,

6}

,

b = {1,

0,

1} ,

c = {2,

−6,

17}.

5.12. a = {6,

3,

4} ,

b = {−1,

−2, −1} ,

c =

{2,

1,

2}.

5.13. a = {7,

3,

4} ,

b = {−1,

−2, −1} ,

c =

{4,

2,

4}.

5.14. a = {2,

3,

2} ,

b = {4,

7,

5} ,

c = {2,

0,

−1}.

5.15. a = {5,

3,

4} ,

b = {−1,

0, −1} ,

c = {4,

2,

4}.

5.16. a = {3,

10,

5} ,

b = {−2,

−2,

−3}

,

c = {2,

4,

3}.

5.17. a = {−2, −4,

−3} , b = {4,

3,

1} ,

c = {6,

7,

4}.

5.18. a = {3,

1,

−1} ,

b = {1,

0,

−1} ,

c = {8,

3,

−2}.

5.19. a = {4,

2,

2} ,

b = {−3,

−3, −3} ,

c = {2,

1,

2}.

5.20. a = {4,

1,

2} ,

b = {9,

2,

5} ,

c = {1,

1,

−1}.

5.21. a = {5,

3,

4} ,

b = {4,

3,

3} ,

c = {9,

5,

8}.

5.22. a = {3,

4,

2} ,

b = {1,

1,

0} ,

c = {8,

11,

6}.

5.23. a = {4,

−1, −6} ,

b = {1,

−3,

−7} ,

c = {2,

−1,

−4}.

5.24. a = {3,

1,

0} ,

b = {−5,

−4, −5} ,

c = {4,

2,

4}.

5.25. a = {3,

0,

3} ,

b = {8,

1,

6} ,

c = {1,

1,

−1}.

229