Учебное пособие 800125
.pdf16.a ={8, 0, 5},b ={2, 0, 1},c ={1, 1, 0},d ={27, 1, 16}.
17.a ={3, 1, 8}, b={0, 1, 3}, c={1, 2, -1}, d={10, 6, 26}.
18.a ={8, 1, 12}, b={1, 2, -1}, c={3, 0, 2}, d={12, 3, 13}.
19.a ={1, -1, -3}, b={1, 4, 1}, c={-3, 2, 1}, d={-3, 11, 1}.
20.a ={1, 7, 10}, b={3, 4 ,5}, c = {4, 3, 4}, d= {0, 8, 11}.
21.a ={-1, 5, 6}, b={0, 5, 1}, c={3, 2, -1}, d={2, 12, 6}
22.a ={1, 0, -1}, b={1, 3, 0}, c={2, -1, 1}, d={2, 2, 2}.
23.a ={2, -4, -3}, b={2, 1, 4}, c={1,-1, 0}, d={7, -8, -2}.
24.a ={-2, 4, 3}, b={2, 1, 4}, c={1,-1, 0}, d={7, -8, -2}.
25.a ={-1, 7, 0}, b={1, -1, 2}, c={3, 2, 0}, d={3, 8, 2}.
26.a ={1, -7, 0}, b={1, -1, 2}, c={3, 2, 0}, d={5, -6, 2}.
27.a ={3, 1, 7}, b={5, 1, 0}, c={2, -4, 1}, d={6, 6, 6}.
28.a ={0, -6, 7}, b={0, -2, 1}, c={3, 1, -1}, d={-3, -9, 9}.
29.a ={1, 1, 9}, b={0, 4, 1}, c={3, -1, 1}, d={4, 4, 11}.
30.a ={1, 1, 8}, b={0, 3, 1}, c={3, 0, 1}, d={4, 4, 10}.
Задание 5. Коллинеарны ли векторы с1 и с2, построенные по векторам a и b?
1.a={1,-2,3},
2.a={1,0,1},
3.a={-2,4,1},
4.a={1,2,-3},
5.a={3,5,4},
6.a={1,4,-2},
7.a={1,-2,5},
8.a={3,4,-1},
9.a={-2,-3,-2}
10.a={-1,4,2},
11.a={5,0,-1},
12.a={0,3,-2},
b={3,0,-1}, b={-2,3,5}, b={1,-2,7}, b={2,-1,-1}, b={5,9,7}, b={1,1,-1}, b={3,-1,0}, b={2,-1,1},
b={1,0,5}, b={3,-2,6}, b={7,2,3}, b={1,-2,1},
c1=2a+4b, c1=a+2b, c1=5a+3b, c1=4a+3b, c1=-2a+b, c1=a+b, c1=4a-2b, c1=6a-3b,
c1=3a+9b, c1=2a-b, c1=2a-b, c1=5a-2b,
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c2=3b-a. c2=3a-b. c2=2a-b. c2=8a-b. c2=3a-2b. c2=4a+2b. c2= b-2a. c2=b-2a.
c2=3b-6a. c2=3b-6a. c2=3b-6a. c2=3a+5b.
13.a={-2,7,-1}, b={-3,5,2},
14.a={3,7,0}, b={1,-3,4},
15.a={-1,2,-1}, b={2,-7,1},
16.a={7,9,-2}, b={5,4,3},
17.a={5,0,-2}, b={6,4,3},
18.a={8,3,-1}, b={4,1,3},
19.a={3,-1 6}, b={5,7,10},
20.a={1,-2,4}, b={7,3,5},
21.a={3,7,0}, b={4,6,-1},
22. a {2,-1,4}, b={3,-7,-6},
23.a={5,-1,-2}, b={6,0,7},
24.a={-9,5,3}, b={7,1,-2},
25.a={4,2,9}, b={0,-1,3},
26.a={2,-1,6}, b={-1,3,8},
27.a={5,0,8}, b={-3,1,7},
28.a={-1,3,4}, b={2,-1,0},
29.a{4,2,-7}, b={5,0,-3},
30.a={2,0,-5}, b={1,-3,4},
c1=2a+3b, c1=4a-2b, c1=6a-2b, c1=4a-b, c1=5a-3b, c1=2a-b, c1=4a-2b, c1=6a-3b, c1=3a+2b, c1=2a-3b, c1=3a-2b, c1=2a-b, c1=4b-3a, c1=5a-2b, c1=3a-4b, c1=6a-2b, c1=a-3b, c1=2a-5b,
c2=3a+2b. c2=b-2a. c2=b-3a. c2=4b-a. c2=6b-10a. c2=2b-4a. c2=b-2a. c2=b-2a. c2=5a-7b. c2=3a-2b. c2=4b-6a. c2=3a+5b. c2=4a-3b. c2=2a-5b. c2=12b-9a. c2=b-3a. c2=6b-2a. c2=5a-2b.
Задание 6. Даны точки А, В, С. Найти косинус угла
между векторами AC и BC . |
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1. |
A (1,-2,3), |
B (0,-1,2), |
C (3,-4,5). |
2. |
А (0,-3,6), |
В (-12,-3,-3), |
С (-9,-3,-6). |
3. |
А (3,3,-1), |
В (5,5,-2), |
С (4,1,1). |
4. |
А (-1,2,-3), |
В (3,4,-6), |
С (1,1,-1). |
5. |
А (-4,-2,0), |
В (-1,-2,4), |
С (3,-2,1). |
6. |
А (5,3,-1), |
В (5,2,0), |
С (6,4,-1). |
7. |
А (-3,-7,-5), |
В (0,-1,-2), |
С (2,3,0). |
8. |
А ( 2,-4,6), |
В (0,-2,4 |
С (6,-8,10). |
9. |
А (0,1,-2), |
В (3,1,2), |
С (4,1,1). |
10. |
А (3,3,-1), |
В (1,5,-2), |
С (4,1,1). |
11. .А (2,1,-1), |
В (6,-1,-4), |
С (4,2,1). |
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12. |
А (-1,-2,1), |
В (-4,-2,5), |
С (-8,-2,2). |
13. |
А (6,2,-3), |
В (6,3,-2), |
С (7,3,-3). |
14. |
А (0,0,4), |
В (-3,-6,1), |
С (-5,-10,-1). |
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10 |
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15. |
А (2,-8,-1), |
В (4,-6,0), |
С (-2,-5,-1). |
16. |
А (3,-6,9), |
В (0,-3,6), |
С (9,-12,15). |
17. |
А (0,2,-4), |
В (8,2,2), |
С (6,2,4). |
18. |
А (3,3,-1), |
В (5,1,-2), |
С (4,1,1). |
19. |
А (-4,3,0), |
В (0,1,3), |
С (-2,4,-2). |
20. |
А (1,-1,0), |
В (-2,-1,4), |
С (8,-1,-1). |
21. |
А (7,0,2), |
В (7,1,3), |
С (8,-1,2). |
22. |
А (2,3,2), |
В (-1,-3,-1), |
С (-3,-7,-3). |
23. |
А (2,2,7), |
В (0,0,6), |
С (-2,5,7). |
24. |
А (-1,2,-3), |
В (0,1,-2), |
С (-3,4,-5). |
25. |
А (0,3,-6), |
В (9,3,6), |
С (12,3,3). |
26. |
А (3,3,-1), |
В (5,1,-2), |
С (4,1,-3). |
27. |
А (-2,1,1), |
В (2,3,-2), |
С (0,0,3). |
28. |
А (1,4,-1), |
В (-2,4,-5), |
С (8,4,0). |
29. |
А (0,1,0), |
В (0,2,1), |
С (1,2,0). |
30. |
А (-4,0,4), |
В (0,2,-4), |
С (-6,8,-10). |
Задание 7. Вычислить площадь параллелограмма, построенного на векторах a и b.
1.a=p+2q;
2.a=3p+q;
3.a=p-3q;
4.a=3p-2q;
5.a=p-2q;
6.a=p+3q;
7.a=2p-q;
8.a=4p+q;
9.a=p-4q;
10.a=p+4q;
11.a=3p+2q;
12.a=4p-q;
13.a=2p+3q;
b=3p-q; |
|p|=1; |
|q|=2; |
b=p-2q; |
|p|=4; |
|q|=1; |
b=p+2q; |
|p|=1/5; |q|=1; |
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b=p+5q; |
|p|=4; |
|q|=1/2; |
b=2p+q; |
|p|=2; |
|q|=3; |
b=p-2q; |
|p|=2; |
|q|=3; |
b=p+3q; |
|p|=3; |
|q|=2; |
b=p-q; |
|p|=7; |
|q|=2; |
b=3p+q; |
|p|=1; |
|q|=2; |
b=2p-q; |
|p|=7; |
|q|=2; |
b=p-q; |
|p|=10; |
|q|=1; |
b=p+2q; |
|p|=5; |
|q|=4; |
b=p-2q; |
|p|=6; |
|q|=7; |
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(p^q)= / 6 ; (p^q)= / 4 ; (p^q)= / 2 ; (p^q)= 5 / 6 ; (p^q)= 3 / 4 ; (p^q)= / 3; (p^q)= / 2 ; (p^q)= / 4 ; (p^q)= / 6 ; (p^q)= / 3; (p^q)= / 2 ; (p^q)= / 4 ; (p^q)= / 3;
14. |
a=3p-q; |
b=p+2q; |
|p|=3; |
|q|=4; |
(p^q)= / 3; |
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15. |
a=2p+3q; |
b=p-2q; |
|p|=2; |
|q|=3; |
(p^q)= / 4 ; |
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16. |
a=2p-3q; |
b=3p+q; |
|p|=4; |
|q|=1; |
(p^q)= / 6 ; |
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17. |
a=5p+q; |
b=p-3q; |
|p|=1; |
|q|=2; |
(p^q)= / 3; |
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18. |
a=7p-2q; |
b=p+3q; |
|p|=1/2; |q|=2; |
(p^q)= / 2 ; |
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19. |
a=6p-q; |
b=p+q; |
|p|=3; |
|q|=4; |
(p^q)= / 4 ; |
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20. |
a=10p+q; |
b=3p-2q; |p|=4; |
|q|=1; |
(p^q)= / 6 ; |
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21. |
a=6p-q; |
b=p+2q; |
|p|=8; |
|q|=1/2; (p^q)= / 3; |
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22. |
a=3p+4q; |
b=q-p; |
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|p|=2,5 |
|q|=2; |
(p^q)= / 2 ; |
23. |
a=7p+q; |
b=p-3q; |
|p|=3; |
|q|=1; |
(p^q)= 3 / 4 ; |
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24. |
a=p+3q; |
b=3p-q; |
|p|=3; |
|q|=5; |
(p^q)= 2 / 3; |
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25. |
a=3p+q; |
b=p-3q; |
|p|=7; |
|q|=2; |
(p^q)= / 4 ; |
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26. |
a=5p-q; |
b=p+q; |
|p|=5; |
|q|=3; |
(p^q)= 5 / 6 ; |
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27. |
a=3p-4q; |
b=p+3q; |
|p|=2; |
|q|=3; |
(p^q)= / 4 ; |
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28. |
a=6p-q; |
b=5q+p; |
|p|=1/2 |
|q|=4; |
(p^q)= 5 / 6 ; |
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29. |
a=2p+3q; |
b=p-2q; |
|p|=2; |
|q|=1; |
(p^q)= / 3; |
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30. |
a=2p-3q; |
b=5p+q; |
|p|=2; |
|q|=3; |
(p^q)= / 2 ; |
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Задание 8. Компланарны ли векторы a, b и c. |
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1. |
a={2,3,1}; |
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b={-1,0,-1}; |
c={2,2,2}. |
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2. |
a={3,2,1}; |
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b={2,3,4}; |
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c={3,1,-1}. |
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3. |
a={1,5,2}; |
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b={-1,1,-1}; |
c={1,1,1}. |
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4. |
a={1,-1,-3}; |
b={3,2,1}; |
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c={2,3,4}. |
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5. |
a={3,3,1}; |
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b={1,-2,1}; |
c={1,1,1}. |
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6. |
a={3,1,-1}; |
b={-2,-1,0}; |
c={5,2,-1}. |
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7. |
a={4,3,1}; |
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b={1,-2,1}; |
c={2,2,2}. |
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8. |
a={4,3,1}; |
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b={6,7,4}; |
c={2,0,-1}. |
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9. |
a={3,2,1}; |
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b={1,-3,-7}; |
c={1,2,3}. |
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10. |
a={3,7,2}; |
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b={-2,0,-1}; |
c={2,2,1}. |
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11. |
a={1,-2,6}; |
b={1,0,1}; |
c={2,-6,17}. |
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12 |
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12. |
a={6,3,4}; |
b={-1,-2,-1}; |
13. |
a={7,3,4}; |
b={-1,-2,-1}; |
14. |
a={2,3,2}; |
b={4,7,5}; |
15. |
a={5,3,4}; |
b={-1,0,-1}; |
16. |
a={3,10,5}; |
b={-2,-2,-3}; |
17. |
a={-2,-4,-3}; |
b={4,3,1}; |
18. |
a={3,1,-1}; |
b={1,0,-1}; |
19. |
a={4,2,2}; |
b={-3,-3,-3}; |
20. |
a={4,1,2}; |
b={9,2,5}; |
21. |
a={5,3,4}; |
b={4,3,3}; |
22. |
a={3,4,2}; |
b={1,1,0}; |
23. |
a={4,-1,-6}; |
b={1,-3,-7}; |
24. |
a={3,1,0}; |
b={-5,-4,-5}; |
25. |
a={3,0,3}; |
b={8,1,6}; |
26. |
a={1,-1,4}; |
b={1,0,3}; |
27. |
a={6,3,4}; |
b={-1,-2,-1}; |
28. |
a={4,1,1}; |
b={-9,-4,-9}; |
29. |
a={-3,3,3}; |
b={-4,7,6}; |
30. |
a={-7,-10,-5}; |
b={0,-2,-1}; |
c={2,1,2}. c={4,2,4}. c={2,0,-1}. c={4,2,4}. c={2,4,3}. c={6,7,4}. c={8,3,-2}. c={2,1,2}. c={1,1,-1}. c={9,5,8}. c={8,11,6}. c={2,-1,-4}. c={4,2,4}. c={1,1,-1}. c={1,-3,-8}. c={2,1,2}. c={6,2,6}. c={3,0,-1}. c={-2,4,-1}.
Задание 9. Даны точки А1, А2, А3 и А4. Найти длину отрезка А1А2, площадь треугольника А1А2А3, длину высоты A1 H треугольника А1А2А3, длину медианы А1 M треугольника
А1А2А3 ,координаты точки K , делящей отрезок А2А3 в отношении 1:2, вычислить объем тетраэдра А1А2А3 А4 и его высоту, опущенную из вершины А1 на грань А1А2А3.
1. |
A1(1,3,6), |
A2(2,2,1), |
A3(-1,0,1), |
A4(-4,6,-3). |
2. |
A1(-4,2,6), |
A2(2,-3,0), |
A3(-10,5,8), |
A4(-5,2,-4). |
3. |
A1(7,2,4), |
A2(7,-1,-2), |
A3(3,3,1), |
A4(-4,2,1). |
4. |
A1(2,1,4), |
A2(-1,5,-2), |
A3(-7,-3,2), |
A4(-6,-3,6). |
5. |
A1(-1,-5,2), A2(-6,0,-3), |
A3(3,6,-3), |
A4(-10,6,7). |
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6. |
A1(0,-1,-1), A2(-2,3,5), |
A3(1,-5,-9), |
A4(-1,-6,3). |
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7. |
A1(5,2,0), |
A2(2,5,0), |
A3(1,2,4), |
A4(-1,1,1). |
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13 |
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8.A1(2,-1,-2), A2(1,2,1),
9.A1(-2,0,-4), A2(-1,7,1),
10.A1(14,4,5), A2(-5,-3,2),
11.A1(1,2,0), A2(3,0,-3),
12.A1(2,-1,2), A2(1,2,-1),
13.A1(1,1,2), A2(-1,1,3),
14.A1(2,3,1), A2(4,1,-2),
15.A1(1,1,-1), A2(2,3,1),
16.A1(1,5,-7), A2(-3,6,3),
17.A1(-3,4,-7), A2(1,5,-4),
18.A1(-1,2,-3), A2(4,-1,0),
19.A1(4,-1,3), A2(-2,1,0),
20.A1(1,-1,1), A2(-2,0,3),
21.A1(1,2,0), A2(1,-1,2),
22.A1(1,0,2), A2(1,2,-1),
23.A1(1,2,-3), A2(1,0,1),
24.A1(3,10,-1), A2(-2,3,-5),
25.A1(-1,2,4), A2(-1,-2,-4),
26.A1(0,-3,1), A2(-4,1,2),
27.A1(1,3,0), A2(4,-1,2),
28.A1(-2,-1,-1), A2(0,3,2),
29.A1(-3,-5,6), A2(2,1,-4),
30.A1(2,-4,-3), A2(5,-6,0),
A3(5,0,-6), A3(4,-8,-4), A3(-2,-6,-3), A3(5,2,6), A3(3,2,1), A3(2,-2,4), A3(6,3,7), A3(3,2,1), A3(-2,7,3),
A3(-5,-2,0), A3(2,1,-2), A3(0,-5,1), A3(2,1,-1), A3(0,1,-1), A3(2,-2,1), A3(-2,-1,6), A3(-6,0,-3), A3(3,0,-1), A3(2,-1,5), A3(3,0,1), A3(3,1,-4), A3(0,-3,-1), A3(-1,3,-3),
A4(-10,9,-7). A4(1,-4,6). A4(-2,2,-1). A4(8,4,-9). A4(-4,2,5). A4(-1,0,-2). A4(7,5,-3). A4(5,9,-8).
A4(-4,8,-12). A4(2,5,4). A4(3,4,5). A4(3,2,-6). A4(2,-2,-4). A4(-3,0,1). A4(2,1,0). A4(0,-5,-4). A4(1,-1,2). A4(7,-3,1). A4(3,1,-4). A4(-4,3,5). A4(-4,7,3). A4(-5,2,-8). A4(-10,-8,7).
Задание 10. Даны уравнения двух плоскостей П1 , П2 и координаты точки M 0 . Найти угол между плоскостями, отрезки, отсекаемые плоскостью П1 на координатных осях, уравнение плоскости, параллельной плоскости П2 , и проходящей через точку M 0 , канонические уравнения
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прямой, являющейся линией пересечения плоскостей П1 , П2 , уравнение плоскости, проходящей через точку M 0 и линию пересечения плоскостей П1 , П2 .
1.3x+2y+z-6=0, 5x+y-z-7=0, M0 (1,2,3) .
2.x-2y+3z-1=0, 2x-y-3=0, M0 (3, 1,1) .
3.2x-3y-z-5=0, x+4y+z-3=0, M0 (2, 2, 2) .
4.x+4y-2z-1=0, 5x+y-z-2=0, M0 (3,1,0) .
5.3x-2y-4=0, 2x+y-z-5=0, M0 (0,1,3) .
6.2x+y-5z-3=0, 3x-y+2z-5=0, M0 (2,5, 1) .
7.7x+y-z-7=0, 2x-3y+z-6=0, M0 (3,0,2) .
8.8x-5y-8=0, 4x+3y-z-7=0, M0 (2, 1,0) .
9.2x+3y-z-5=0, 3x-y+z-2=0, M0 (1,3,3) .
10.6x+2y+z-3=0, x-y+5z-4=0, M0 (2,3,2) .
11.4y+3z-7=0, x+2y+3z-5=0, M0 (1, 2,3) .
12.5x+2y-3=0, 2x+4y-z+2=0, M0 (1,3,4) .
13.7x-y+z-6=0, 3x+2y-5=0, M0 (0,4,2) .
14.5x+3y-2z-2=0, 3x-y+4z-4=0, M0 (1, 3,0) .
15.3x+5y-z-2=0, x-y+4z+2=0, M0 (3,1, 1) .
16.3x-3y+z-2=0, 2x+5y-4z+1=0, M0 (1,0,3) .
17.4x-y+z-4=0, 2x+3y-z-4=0, M0 (2, 2,1) .
18.5x-y+2z-4=0, 3x+y+2z-2=0, M0 (1, 1,0) .
19.3x-2y+z-2=0, 3x+2y-z-4=0, M0 (2,1, 2) .
20.x+y+3z-3=0, 3x+z-7=0, M0 (1,3,3) .
21.2x+y-z-2=0, 3x+y-1=0, M0 (0,3,4) .
22.3x+5y-z-6=0, x+y-2z+2=0, M0 (2,3,2) .
23.2x-2y+z-3=0, 3x+y-3z-4=0, M0 (1, 1,1) .
24.3x+y-2z-4=0, 2x-y+z-4=0, M0 (3,4,3) .
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25.4x-z-3=0, 2x+y+2z-1=0, M0 (2, 3, 2) .
26.2x+y+3z-4=0, 2y+z-7=0, M0 (1,2,3) .
27.5x-y+z-5=0, 2x+z-3=0, M0 (2,5, 1) .
28.3x-y+2z-5=0, 3x-y-2=0, M0 (1, 2,0) .
29.4x+y-2z+1=0, 3x+y-z=0, M0 (2,3, 1) .
30.3x+3y-z-1=0, 2x+y+z-1=0, M0 (2, 1,2) .
Задание 11. Даны две прямые l1 , l2 и плоскость П . Найти угол между прямой l2 и плоскостью П , уравнение плоскости, проходящей через прямую l1 параллельно прямой l2 , координаты точки пересечения прямой l1 и плоскости П .
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Задание 12. Привести общее уравнение кривой каноническому виду и построить полученную кривую.
1.x2 – 3y2 + x – 4y +2 = 0;
2.2x2 + y2 –4y – 1 =0;
3.x2 – 2x + 5y + 2 = 0;
4.x2 + 4y2 – 6x - 1 = 0;
5.3x2 – x – y2 –2у + 4 = 0;
6.x2 – x + 7y + 2 = 0;
7.x2 + х + 3y2 – 9y – 1 = 0;
8.2x2 - 6x – 3y2 – 5 = 0;
9.x2 + y2 – 4y = 0;
10.4x2 + y2 – y = 0;
11.x2 + 8x – y + 5 = 0;
12.x2 + y2 + x – y – 10 = 0;
13.x + 2y2 – y +4 = 0;
14.x2 – 3x – y2 + 4y + 5 =0;
15.3x2 -12x+ 4y2 – 2 = 0;
18