Добавил:

Vezen Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Коломенский институт филиал МГМУ

Предмет:

Высшая математика

Файл:

Кузнецов_математика

.pdf

Скачиваний:

408

Добавлен:

20.06.2014

Размер:

2.87 Mб

Скачать

☆

<<< < Предыдущая 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2223 / 2723 24 25 26 27 > Следующая >>>

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.
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.

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,

−2),

C(4,

1).

3.4. A(−1,

−3),

B(3,

4, −6),

C(1,

−1).

3.5. A(−4,

−2, 0),

B(−1,

−2,

4), C(3,

−2,

1).

3.6. A(5,

−1),

B(5,

0),

C(6,

−1).

3.7. A(−3,

−7, −5),

B(0,

−1,

−2),

C(2,

0).

3.8. A(2,

−4, 6),

B(0,

−2,

−8,

10).

3.9. A(0,

1, −2),

B(3,

2),

(4,

1).

3.10. A(3,

3, −1),

B(1,

−2),

C(4,

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,

−2),

C(7,

−3).

3.14. A(0,

4),

B(−3,

−6, 1),

C(−5,

−10,

−1).

3.15. A(2,

−8,

−1),

B(4,

−6,

, 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).

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

4.1. a = p + 2q,

b = 3p − q;

=1,

= 2,

(p q)= π

4.2. a = 3p + q,

b = p − 2q;

= 4,

=1,

(p q)= π

4.3. a = p − 3q,

b = p + 2q;

=1 5,

=1, (p q)= π 2.

4.4. a = 3p − 2q, b = p + 5q;

= 4,

=1 2, (p q)= 5π 6.

4.5. a = p − 2q,

b = 2p + q;

= 2,

= 3,

(p q)= 3π 4.

4.6. a = p + 3q,

b = p − 2q;

= 2,

= 3,

(p q)= π 3.

4.7. a = 2p − q,

b = p + 3q;

= 3,

= 2,

(p q)= π 2.

4.8. a = 4p + q,

b = p − q;

= 7,

= 2,

(p q)= π 4.

4.9. a = p − 4q,

b = 3p + q;

=1,

= 2,

(p q)= π 6.

4.10. a = p + 4q,

b = 2p − q;

= 7,

= 2,

(p q)= π 3.

4.11. a = 3p + 2q,

b = p − q;

=10,

=1,

(p q)= π 2.

4.12. a = 4p − q,

b = p + 2q;

= 5,

= 4,

(p q)= π 4.

4.13. a = 2p + 3q,

b = p − 2q;

= 6,

= 7,

(p q)= π 3.

4.14. a = 3p − q,

b = p + 2q;

= 3,

= 4,

(p q)= π 3.

4.15. a = 2p + 3q,

b = p − 2q;

= 2,

= 3,

(p q)= π 4.

4.16. a = 2p − 3q,

b = 3p + q;

= 4,

=1,

(p q)= π 6.

4.17. a = 5p + q,

b = p − 3q;

=1,

= 2,

(p q)= π 3.

4.18. a = 7p − 2q,

b = p + 3q;

=1 2,

= 2,

(p q)= π 2.

4.19. a = 6p − q,

b = p + q;

= 3,

= 4, (p q)= π 4.

4.20. a =10p + q,

b = 3p − 2q;

= 4,

=1,

(p q)= π 6.

4.21. a = 6p − q,

b = p + 2q;

= 8,

=1 2,

(p q)= π 3.

4.22. a = 3p + 4q,

b = q − p;

= 2,5,

= 2, (p q)= π 2.

4.23. a = 7p + q,

b = p − 3q;

= 3,

=1,

(p q)= 3π 4.

4.24. a = p + 3q,

b = 3p − q;

= 3,

= 5,

(p q)= 2π 3.

4.25. a = 3p + q,

b = p − 3q;

= 7,

= 2,

(p q)= π 4.

4.26. a = 5p − q,

b = p + q;

= 5,

= 3, (p q)= 5π 6.

4.27. a = 3p − 4q,

b = p + 3q;

= 2,

= 3,

(p q)= π 4.

4.28. a = 6p − q,

b = 5q + p;

=1 2,

= 4, (p q)= 5π 6.

4.29. a = 2p + 3q,

b = p − 2q;

= 2,

=1,

(p q)= π 3.

4.30. a = 2p − 3q,

b = 5p + q;

= 2,

= 3,

(p q)= π 2.

4.31. a = 3p + 2q,

b = 2p − q;

= 4,

= 3,

(p q)= 3π 4.

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

5.1. a = {2,

1},

b = {−1,

−1},

c = {2,

2}.

5.2. a = {3,

1},

b = {2,

4},

c = {3,

−1}.

5.3. a = {1,

2},

b = {−1,

−1},

c = {1,

1}.

5.4. a = {1,

−1,

−3}, b = {3,

1},

c = {2,

4}.

5.5. a = {3,

1},

b = {1,

−2,

1},

c = {1,

1}.

5.6. a = {3,

−1},

b = {−2,

−1,

0},

c = {5,

−1}.

5.7. a = {4,

1},

b = {1,

−2,

1},

c = {2,

2}.

5.8. a = {4,

1},

b = {6,

4},

c = {2,

−1}.

5.9. a = {3, 2, 1}, b = {1, −3,

−7},

c = {1,

3}.

5.10. a = {3,

2},

b = {−2,

−1}

c = {2,

1}.

5.11. a = {1,

−2,

6},

b = {1,

1},

c = {2,

−6,

17}.

5.12. a = {6,

4},

b = {−1,

−2,

−1},

c = {2,

2}.

5.13. a = {7,

4},

b = {−1,

−2,

−1},

c = {4,

4}.

5.14. a = {2,

2},

b = {4,

5},

c = {2,

−1}.

5.15. a = {5,

4},

b = {−1,

−1},

c = {4,

4}.

5.16. a = {3,

10,

5},

b = {−2,

−2,

−3}

c = {2,

4, 3}.

5.17. a = {−2,

−4, −3},

b = {4,

1},

c = {6,

4}.

5.18. a = {3,

−1},

b = {1,

−1},

c = {8,

−2}.

5.19. a = {4,

2},

b = {−3,

−3,

−3},

c = {2,

2}.

5.20. a = {4,

2},

b = {9,

5},

c = {1,

1, −1}.

5.21. a = {5,

4},

b = {4,

3},

c = {9,

8}.

5.22. a = {3,

2},

b = {1,

0},

c = {8,

11,

6}.

5.23. a = {4,

−1,

−6},

b = {1,

−3,

−7},

c = {2,

−1, −4}.

5.24. a = {3,

0},

b = {−5,

−4,

−5},

c = {4,

4}.

5.25. a = {3,

3},

b = {8,

6},

c = {1,

−1}.

5.26. a = {1,

−1,

4},

b = {1,

c = {1,

−3,

8}.

5.27. a = {6,

4},

b = {−1,

−2,

−1},

c = {2,

2}.

5.28. a = {4,

1},

b = {−9,

−4, −9},

c = {6,

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).

6.16. A1 (1, 5, −7), A2 (−3, 6, 3), A3 (−2, 7, 3), A4 (−4, 8, −12).

6.17. A1 (−3,

−7), A2 (1,

−4),

A3 (−5,

−2,

0), A4 (2,

4).

6.18. A1 (−1,

−3), A2 (4,

−1,

0), A3 (2,

−2),

A4 (3,

5).

6.19. A1 (4,

−1,

3),

A2 (−2, 1,

0),

A3 (0,

−5,

1),

A4 (3,

−6).

6.20. A1 (1,

−1,

1),

A2 (−2, 0,

3),

A3 (2,

−1), A4 (2,

−2,

−4).

6.21. A1 (1,

0),

A2 (1, −1,

2),

A3 (0, 1,

−1),

A4 (−3, 0,

1).

6.22. A1 (1,

2),

A2 (1, 2,

−1),

A3 (2, −2,

1),

A4 (2,

0).

6.23. A1 (1,

−3),

A2 (1,

1),

A3 (−2, −1,

6),

A4 (0,

−5,

−4).

6.24. A1 (3,

10,

−1), A2 (−2,

3, −5), A3 (−6,

−3), A4 (1,

−1,

2).

6.25. A1 (−1,

4), A2 (−1,

−2,

−4), A3 (3,

−1), A4 (7,

−3, 1).

6.26. A1 (0,

−3,

1),

A2 (−4, 1,

2),

A3 (2,

−1,

5),

A4 (3,

−4).

6.27. A1 (1,

0),

A2 (4, −1,

2),

A3 (3,

1),

A4 (−4,

5).

6.28. A1

(−2,

−1, −1),

A2 (0,

2), A3 (3,

−4),

A4 (−4,

3).

6.29. A1

(−3,

−5, 6),

A2 (2,

−4),

A3 (0,

−3,

−1), A4 (−5,

−8).

6.30. A1

(2,

−4,

−3), A2 (5,

−6,

0), A3 (−1,

−3), A4 (−10,

−8, 7).

6.31. A1

(1,

−1,

2),

A2 (2,

2),

A3 (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).
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),		M	3 (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).
7.30. M1	(1,	1,	−1),	M2 (2,		3,		1),	M	3 (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).
8.20.	A(1,	0, −6),		B(−7,		2,	1), C(−9,					6,		1).
8.21.	A(−3,	1,	0),	B(6,	3,		3), C(9,				4,		−2).
8.22.	A(−4,	−2, 5),		B(3,		−3, −7),				C(9,			3, −7).
8.23.	A(0,	−8, 10),		B(−5, 5,				7),	C(−8,				0,		4).
8.24.	A(1,	−5, −2),		B(6,		−2, 1),			C(2,			−2, −2).
8.25.	A(0,	7,	−9),	B(−1,		8,	−11), C(−4,						3,		−12).
8.26.	A(−3,	−1, 7),		B(0,		2,		−6), C(2,				3, −5).
8.27.	A(5,	3, −1),		B(0,	0,		−3),		C(5, −1,					0).
8.28.	A(−1,	2,	−2),	B(13,		14,		1),	C(14,				15,		2).
8.29.	A(7,	−5,	0),	B(8,	3,		−1),		C(8,		5,		1).

<<< < Предыдущая 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2223 / 2723 24 25 26 27 > Следующая >>>

Соседние файлы в предмете Высшая математика

#
20.06.2014380.9 Кб75Доказательства по линейной алгебре.docx
#
20.06.20142.87 Mб408Кузнецов_математика.pdf
#
20.06.2014951.81 Кб526Ответы на вопросы к экзамену по математике (шпаргалки).doc
#
20.06.20142.31 Mб175ТПУ Линейная алгебра 1курс ИДЗ №1.doc
#
20.06.2014380.42 Кб64ТПУ Линейная алгебра 1курс ИДЗ №2.doc
#
20.06.2014539.14 Кб56ТПУ Линейная алгебра 1курс ИДЗ №3.doc
#
20.06.20141.09 Mб65ТПУ Линейная алгебра 1курс ИДЗ №4.doc