5.26. a = {1,

−1, 4} ,

b = {1,

0,

3}

,

c =

{1,

−3,

8}.

5.27. a = {6,

3,

4} ,

b = {−1,

−2,

−1} ,

c = {2,

1,

2}.

5.28. a = {4,

1,

1} ,

b = {−9,

−4, −9} ,

c = {6,

2,

6}.

5.29. a = {−3,

3, 3} ,

b = {−4,

7,

6} , c = {3,

0,

−1}.

5.30. a = {−7,

10,

−5} ,

b = {0,

−2,

−1} ,

c = {−2,

4, −1}.

5.31. a = {7,

4,

6} ,

b = {2,

1,

1} ,

c = {19,

11,

17}.

Задача 6. Вычислить объем тетраэдра с вершинами в точках A1,

A2 ,

A3 ,

A4 и его

высоту, опущенную из вершины A4 на грань A1A2 A3 .

6.1. A1 (1, 3,

6),

A2 (2,

2,

1), A3 (−1, 0,

1),

A4 (−4,

6,

−3).

6.2. A1 (−4,

2,

6),

A2 (2, −3,

0), A3 (−10,

5,

8), A4 (−5,

2, −4).

6.3. A1 (7,

2,

4),

A2 (7, −1,

−2),

A3 (3, 3,

1),

A4 (−4,

2,

1).

6.4. A1 (2,

1,

4),

A2 (−1, 5,

−2), A3 (−7,

−3,

2), A4 (−6,

−3, 6).

6.5. A1 (−1,

−5, 2),

A2 (−6,

0,

−3), A3 (3,

6,

−3), A4 (−10,

6,

7).

6.6. A1 (0,

−1,

−1), A2 (−2,

3,

5),

A3 (1,

−5,

−9), A4 (−1,

−6,

3).

6.7. A1 (5,

2,

0),

A2 (2, 5,

0), A3 (1, 2,

4),

A4 (−1, 1, 1).

6.8. A1 (2,

−1,

−2),

A2 (1,

2,

1),

A3 (5,

0,

−6),

A4 (−10,

9, −7).

6.9. A1 (−2,

0,

−4), A2 (−1,

7,

1), A3 (4,

−8,

−4), A4 (1,

−4, 6).

6.10. A1 (14,

4,

5), A2 (−5,

−3, 2),

A3 (−2,

−6,

−3), A4 (

−2,

2,

−1).

6.11. A1 (1,

2,

0),

A2 (3,

0,

−3), A3 (5, 2, 6),

A4 (8, 4,

−9).

6.12. A1 (2,

−1,

2)

,

A2 (1, 2,

−1),

A3 (3, 2,

1),

A4 (−4, 2,

5).

6.13. A1 (1,

1,

2),

A2 (−1, 1,

3),

A3 (2, −2,

4),

A4 (−1, 0,

−2).

6.14. A1 (2,

3,

1),

A2 (4,

1,

−2),

A3 (6, 3,

7),

A4 (7, 5,

−3).

6.15. A1 (1,

1,

−1),

A2 (2,

3,

1),

A3 (3,

2,

1),

A4 (5,

9,

−8).

230

6.16. A1 (1,

5,

−7), A2 (−3,

6,

3), A3 (−2,

7,

3),

A4 (−4,

8, −12).

6.17. A1 (−3,

4,

−7), A2 (1,

5,

−4), A3 (−5,

−2,

0), A4 (2,

5,

4).

6.18. A1 (−1,

2,

−3), A2 (4,

−1,

0), A3 (2,

1,

−2), A4 (3,

4,

5).

6.19. A1 (4,

−1,

3), A2 (−2,

1,

0), A3 (0,

−5,

1),

A4 (3, 2,

−6).

6.20. A1 (1,

−1,

1),

A2 (−2, 0,

3),

A3 (2,

1,

−1),

A4 (2, −2,

−4).

6.21. A1 (1,

2,

0),

A2 (1, −1,

2), A3 (0,

1,

−1),

A4 (−3, 0,

1).

6.22. A1 (1,

0,

2),

A2 (1, 2,

−1), A3 (2,

−2,

1),

A4 (2,

1,

0).

6.23. A1 (1,

2,

−3),

A2 (1, 0,

1),

A3 (−2,

−1,

6)

,

A4 (0, −5,

−4).

6.24. A1 (3,

10,

−1), A2 (−2,

3,

−5), A3 (−6,

0,

−3), A4 (1,

−1,

2).

6.25. A1 (−1,

2,

4), A2 (−1,

−2,

−4), A3 (3,

0,

−1), A4 (7,

−3, 1).

6.26. A1 (0,

−3,

1),

A2 (−4, 1,

2), A3 (2,

−1,

5),

A4 (3, 1,

−4).

6.27. A1 (1,

3,

0),

A2 (4, −1,

2),

A3 (3,

0,

1),

A4 (−4, 3,

5).

6.28. A1

(−2,

−1, −1),

A2 (0,

3,

2), A3 (3,

1,

−4),

A4 (−4,

7,

3).

6.29. A1

(−3,

−5, 6),

A2 (2,

1,

−4), A3 (0,

−3,

−1), A4 (−5,

2, −8).

6.30. A1

(2,

−4,

−3), A2 (5,

−6,

0), A3 (−1,

3,

−3), A4 (−10,

−8, 7).

6.31. A1

(1,

−1,

2),

A2 (2, 1,

2),

A3 (1,

1,

4),

A4 (6,

−3,

8).

Задача 7. Найти расстояние от точки M0 до плоскости, проходящей через точки

M1, M2 , M3 .

7.1. M1 (−3,

4,

−7), M2 (1,

5,

−4), M3 (−5,

−2, 0),

M0 (−12,

7, −1).

7.2. M1

(−1,

2,

−3),

M2 (4,

−1,

0), M3 (2,

1, −2),

M0 (1, −6,

−5).

7.3. M1

(−3,

−1, 1),

M2 (−9,

1,

−2), M3 (3,

−5,

4), M0 (−7,

0,

−1).

7.4. M1

(1,

−1,

1), M2 (−2,

0,

3),

M3 (2, 1,

−1),

M0 (−2, 4,

2).

7.5. M1

(1,

2,

0), M2 (1, −1, 2),

M3 (0, 1,

−1),

M0 (2, −1,

4).

231

7.6. M1 (1,	0,		2), M2 (1, 2,					−1), M3 (2,						−2,		1),		M0 (−5,		−9,		1).
7.7. M1 (1,	2,		−3),		M2 (1,		0,		1), M3 (−2,						−1,		6)	, M0 (3,		−2,		−9).
7.8. M1 (3,	10,			−1), M2 (−2,				3,		−5),			M3 (−6,			0,		−3), M0 (−6,			7, −10).
7.9. M1 (−1,		2,		4),	M2 (−1,		−2,				−4), M3 (3,					0,		−1), M0 (−2,				3,	5).
7.10. M1 (0,		−3,		1), M2 (−4,				1,			2),		M3 (2,		−1,		5),		M0 (−3,		4,	−5).
7.11. M1 (1,		3,		0),	M2 (4,		−1,		2),			M3 (3,			0,	1),		M0 (4, 3,			0).
7.12. M1 (−2,		−1, −1),				M2 (0,			3,		2),		M3 (3, 1,				−4), M0 (−21,				20, −16).
7.13. M1 (−3,		−5, 6),				M2 (2,		1,			−4),		M3 (0,			−3,		−1), M0 (3,			6,		68).
7.14. M1 (2,		−4,		−3), M2 (5,				−6,			0),		M3 (−1,			3,		−3), M0 (2,			−10, 8).
7.15. M1 (1,		−1,		2),	M2 (2,		1,		2),			M3 (1,			1,	4),		M0 (−3, 2,			7).
7.16. M1 (1,		3,		6),	M2 (2,		2,		1), M3 (−1,						0,	1),		M0 (5, −4,			5).
7.17. M1 (−4,		2,		6), M2 (2,			−3,				0),		M3 (−10,			5,		8), M0 (−12,				1,	8).
7.18. M1 (7,		2,		4), M2 (7,			−1,			−2),			M3 (−5,			−2,		−1), M0 (10,				1,	8).
7.19. M1 (2,		1,		4),	M2 (3,		5,		−2),			M3 (−7, −3,					2),		M0 (−3,		1,	8).
7.20. M1 (−1,		−5, 2),				M2 (−6,			0,			3),	M3 (3,			6,		−3), M0 (10,			−8, −7).
7.21. M1 (0,		−1,		−1), M2 (−2,					3,		5),		M3 (1,		−5,		−9), M0 (−4,				−13, 6).
7.22. M1 (5,		2,		0),	M2 (2,		5,		0), M3 (1,					2,		4),		M0 (−3, −6,			−8).
7.23. M1 (2,		−1,		−2),		M2 (1,		2,			1),		M3 (5, 0,				−6), M0 (14,				−3, 7).
7.24. M1 (−2,		0,		−4), M2 (−1,					7,		1),		M3 (4,			−8,		−4), M0 (−6,				5,	5).
7.25. M1 (14,		4,		5),	M2 (−5,			−3, 2),					M3 (−2,			−6,		−3), M0 (−1,				−8, 7).
7.26. M1 (1,		2,		0),	M2 (3,		0,		−3), M3 (5,						2,	6),		M0 (−13, −8,				16).
7.27. M1 (2,		−1,		2),	M2 (1,		2,			−1),			M3 (3,		2,		1),		M0 (−5,	3,		7).
7.28. M1 (1,		1,		2), M2 (−1,			1,		3),			M3 (2,			−2,	4), M0 (2,				3,	8).
7.29. M1 (2,		3,		1),	M2 (4,		1,		−2),			M3 (6,			3,	7),		M0 (−5,		−4,		8).

232

7.30. M1 (1,

1,

−1),

M2 (2,

3,

1), M3 (3,

2,

1), M0 (−3, −7, 6).

7.31. M1 (1,

5,

−7),

M2 (−3,

6, 3), M3 (−2,

7, 3), M0 (1, −1, 2).

Задача 8. Написать уравнение плоскости, проходящей через точку A

перпендикулярно вектору BC .

8.1. A(1,

0, −2),

B(2, −1,

3), C (0,

−3,

2).

8.2. A(−1,

3,

4),

B(−1, 5,

0), C (2,

6,

1).

8.3. A(4,

−2,

0),

B(1, −1,

−5), C (−2,

1, −3).

8.4. A(−8,

0,

7),

B(−3, 2,

4), C (−1,

4,

5).

8.5. A(7,

−5,

1),

B(5,

−1,

−3),

C (3,

0,

−4).

8.6. A(−3,

5,

−2),

B(−4, 0,

3), C (−3,

2,

5).

8.7. A(1,

−1,

8),

B(−4,

−3,

10), C (−1,

−1, 7).

8.8. A(−2,

0,

−5),

B(2,

7,

−3), C (1,

10,

−1).

8.9. A(1,

9, −4),

B(5,

7,

1),

C (3,

5,

0).

8.10. A(−7,

0,

3),

B(1, −5,

−4),

C (2,

−3, 0).

8.11. A(0,

−3,

5),

B(−7, 2,

6), C (−3,

2,

4).

8.12. A(5,

−1,

2),

B(2, −4,

3), C (4,

−1,

3).

8.13. A(−3,

7,

2),

B(3,

5,

1), C (4,

5,

3).

8.14. A(0,

−2,

8),

B(4,

3,

2), C (1,

4,

3).

8.15. A(1,

−1,

5),

B(0,

7, 8), C (−1,

3,

8).

8.16. A(−10,

0, 9)

,

B(12,

4,

11), C (8,

5,

15).

8.17. A(3,

−3,

−6)

,

B(1,

9,

−5), C (6,

6,

−4).

8.18. A(2,

1,

7),

B(9,

0,

2),

C (9,

2,

3).

8.19. A(−7,

1,

−4)

,

B(8,

11,

−3),

C (9,

9,

−1).

233