Векторная алгебра. методические указания для организации самостоятельной работы по курсу Высшая математика. Пантелеев И.Н

1.1.

x ={−2,

4,

7},

p ={0,

1,

2},

q ={1, 0, 1}, r ={−1, 2, 4}.

1.2.

x ={6,

12,

−1},

p ={1,

3,

0},

q ={2, −1, 1}, r ={0, −1, 2}.

1.3.

x ={1, −4,

4}, p ={2, 1,

−1},

q ={0, 3, 2}, r ={1, −1, 1}.

1.4.

x ={−9,

5,

5},

p ={4,

1,

1},

q ={2, 0, −3}, r ={−1, 2, 1}.

1.5.

x ={−5,

−5,

5},

p ={−2,

0,

1},

q ={1, 3, −1}, r ={0, 4, 1}.

1.6.

x ={13,

2,

7},

p ={5,

1,

0},

q ={2, −1, 3}, r ={1, 0, −1}.

1.7.

x ={−19,

−1, 7},

p ={0,

1,

1},

q ={−2, 0, 1}, r ={3, 1, 0}.

1.8.

x ={3,

−3,

4},

p ={1,

0,

2},

q ={0, 1, 1}, r ={2, −1, 4}.

1.9.

x ={3,

3,

−1},

p ={3,

1,

0},

q ={−1, 2, 1}, r ={−1, 0, 2}.

1.10. x ={−1,

7,

−4},

p ={−1,

2,

1},

q ={2, 0, 3}, r ={1, 1, −1}.

1.11. x ={6,

5,

−14},

p ={1,

1,

4},

q ={0, −3, 2}, r ={2, 1, −1}.

1.12. x ={6,

−1, 7},

p ={1,

−2, 0},

q ={−1, 1, 3}, r ={1, 0, 4}.

1.13. x ={5,

15,

0},

p ={1,

0,

5},

q ={−1, 3, 2}, r ={0, −1, 1}.

1.14. x ={2,

−1,

11},

p ={1, 1,

0},

q ={0, 1, −2}, r ={1, 0, 3}.

1.15. x ={11,

5,

−3},

p ={1,

0,

2},

q ={−1, 0, 1}, r ={2, 5, −3}.

1.16. x ={8,

0,

5},

p ={2,

0,

1},

q ={1, 1, 0}, r ={4, 1, 2}.

1.17. x ={3,

1,

8},

p ={0,

1,

3},

q ={1, 2, −1}, r ={2, 0, −1}.

1.18. x ={8,

1,

12},

p ={1,

2,

−1},

q ={3, 0, 2}, r ={−1, 1, 1}.

1.19. x ={−9,

−8, −3}, p ={1,

4,

1},

q ={−3, 2, 0}, r ={1, −1, 2}.

1.20. x ={−5,

9,

−13},

p ={0,

1,

−2},

1.21. x ={−15,

5,

6},

p ={0,

5,

1},

q ={3, 2, −1}, r ={−1, 1, 0}.

1.22. x ={8,

9,

4},

p ={1,

0,

1},

q ={0, −2, 1}, r ={1, 3, 0}.

1.23. x ={23,

−14,

−30},

p ={2,

1,

0},

q ={1, −1, 0}, r ={−3, 2, 5}.

1.24. x ={3,

1,

3},

p ={2, 1,

0},

q ={1, 0, 1}, r ={4, 2, 1}.

1.25. x ={−1,

7,

0},

p ={0,

3,

1},

q ={1, −1, 2}, r ={2, −1, 0}.

1.26. x ={11,

−1,

4},

p ={1,

−1,

2},

q ={3, 2, 0}, r ={−1, 1, 1}.

1.27. x ={−13,

2, 18},

p ={1,

1,

4},

q ={−3, 0, 2}, r ={1, 2, −1}.

1.28. x ={0,

−8,

9},

p ={0,

−2,

1},

q ={3, 1, −1}, r ={4, 0, 1}.

1.29. x ={8,

−7,

−13},

p ={0,

1,

5},

q ={3, −1, 2}, r ={−1, 0, 1}.

1.30. x ={2,

7,

5},

p ={1,

0,

1},

q ={1, −2, 0}, r ={0, 3, 1}.

1.31. x ={15,

−20,

−1},

p ={0,

2, 1},

2.1. a ={1,

−2,

3},

b ={3,

0,

−1},

c1 = 2a +4b, c2 =3b −a.

2.2. a ={1,

0,

1},

b ={−2,

3,

5},

c1 =a +2b, c2 =3a −b.

2.3. a ={−2,

4,

1},

b ={1,

−2,

7},

c1 =5a +3b, c2 = 2a −b.

2.4. a ={1,

2,

−3},

b ={2,

−1,

−1},

c1 = 4a +3b, c2 =8a −b.

2.5. a ={3,

5,

4},

b ={5,

9,

7},

c1 = −2a +b, c2 =3a −2b.

2.6. a ={1,

4,

−2},

b ={1,

1,

−1},

c1 =a +b, c2 = 4a +2b.

2.7. a ={1,

−2,

5},

b ={3,

−1,

0},

c1 = 4a −2b, c2 =b −2a.

2.8. a ={3,

4,

−1},

b ={2,

−1,

1},

c1 = 6a −3b, c2 =b −2a.

2.9. a ={−2,

−3, −2}, b ={1,

0, 5},

c1 =3a +9b, c2 = −a −3b.

33

2.10. a ={−1,

4,

2},

b ={3,

−2,

6},

c1 = 2a −b, c2 =3b −6a.

2.11. a ={5,

0,

−1},

b ={7,

2,

3},

c1 = 2a −b, c2 =3b −6a.

2.12. a ={0,

3,

−2},

b ={1,

−2,

1},

c1 = 5a −2b, c2 =3a +5b.

2.13. a ={−2,

7,

−1},

b ={−3,

5,

2},

c1 = 2a +3b, c2 =3a + 2b.

2.14. a ={3,

7,

0}, b ={1,

−3,

4},

c1 = 4a −2b, c2 =b −2a.

2.15. a ={−1,

2,

−1},

b ={2,

−7, 1},

c1 = 6a −2b, c2 =b −3a.

2.16. a ={7,

9,

−2},

b ={5,

4,

3},

c1 = 4a −b, c2 = 4b −a.

2.17. a ={5,

0,

−2},

b ={6,

4,

3},

c1 =5a −3b, c2 = 6b −10a.

2.18. a ={8,

3,

−1},

b ={4,

1,

3},

c1 = 2a −b, c2 = 2b −4a.

2.19. a ={3,

−1,

6},

b ={5,

7,

10},

c1 = 4a −2b, c2 =b −2a.

34

2.20. a ={1,

−2,

4},

b ={7,

3,

5},

c1 = 6a −3b, c2 =b −2a.

2.21. a ={3,

7,

0},

b ={4,

6,

−1},

c1 =3a +2b, c2 = 5a −7b.

2.22. a ={2,

−1,

4},

b ={3,

−7,

−6},

c1 = 2a −3b, c2 =3a −2b.

2.23. a ={5,

−1,

−2},

b ={6,

0,

7},

c1 =3a −2b, c2 = 4b −6a.

2.24. a ={−9,

5,

3},

b ={7,

1,

−2},

c1 = 2a −b, c2 =3a +5b.

2.25. a ={4,

2,

9},

b ={0,

−1,

3},

c1 = 4b −3a, c2 = 4a −3b.

2.26. a ={2,

−1,

6},

b ={−1,

3,

8},

c1 =5a −2b, c2 = 2a −5b.

2.27. a ={5,

0,

8},

b ={−3,

1,

7},

c1 =3a −4b, c2 =12b −9a.

2.28. a ={−1,

3,

4},

b ={2,

−1,

0},

c1 = 6a −2b, c2 =b −3a.

2.29. a ={4,

2, −7},

b ={5,

0,

−3},

c1 =a −3b, c2 = 6b −2a.

35

2.30. a ={2, 0,

−5},

b ={1,

−3,

4},

c1 = 2a −5b, c2 =5a −2b.

2.31. a ={−1,

2,

8},

b ={3,

7,

−1},

c1 = 4a −3b, c2 = 9b −12a.

JG

JJG

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).

3.13. 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).

36

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).

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).

на векторах a и b .

4.1.

(p q)=π 6.

a = p + 2q, b = 3p −q;

p

=1,

q

= 2,

4.2.

=1, (p q)=π 4.

a = 3p +q, b = p −2q;

p

= 4,

q

4.3.

(p q)=π 2.

a = p −3q, b = p + 2q; p =1 5, q =1,

4.4.

(p q)= 5π 6.

a = 3p −2q, b = p +5q; p = 4, q =1 2,

4.5.

(p q)=3π 4.

a =p −2q, b = 2p +q;

p

= 2,

q

=3,

4.6.

(p q)=π 3.

a = p +3q, b = p −2q;

p

= 2,

q

= 3,

4.7.

(p q)=π 2.

a = 2p −q, b = p +3q;

p

= 3,

q

= 2,

4.8.

(p q)=π 4.

a = 4p +q, b = p −q;

p

= 7,

q

= 2,

4.9.

(p q)=π 6.

a = p −4q, b = 3p +q;

p

=1,

q

= 2,

4.10.

(p q)=π 3.

a = p + 4q, b = 2p −q;

p

= 7,

q

= 2,

4.11.

(p q)=π 2.

a = 3p + 2q, b = p −q;

p

=10,

q

=1,

4.12.

(p q)=π 4.

a = 4p −q, b = p + 2q;

p

= 5,

q

= 4,

38

4.13.

(p q)=π 3.

a = 2p +3q, b = p −2q;

p

= 6,

q

= 7,

4.14.

(p q)=π 3.

a = 3p −q, b = p + 2q;

p

= 3,

q

= 4,

4.15.

(p q)=π 4.

a = 2p +3q, b = p −2q;

p

= 2,

q

= 3,

4.16.

(p q)=π 6.

a = 2p −3q, b = 3p +q;

p

= 4,

q

=1,

4.17.

(p q)=π 3.

a = 5p +q, b = p −3q;

p

=1,

q

= 2,

4.18.

a = 6p −q, b = p +q;

p

= 3,

q

= 4,

(p q)=π 4.

4.20.

(p q)=π 6.

a =10p +q, b = 3p −2q;

p

= 4,

q

=1,

4.21.

(p q)=π 3.

a = 6p −q, b = p + 2q; p =8, q =1 2,

4.22.

(p q)=π 2.

a = 3p + 4q, b = q −p;

p

= 2,5,

q

= 2,

4.23.

(p q)= 3π 4.

a = 7p +q, b = p −3q;

p

= 3,

q

=1,

4.24.