Задачи по аналитической геометрии. Часть II. Игудесман К.Б. - 56 стр.

UptoLike

ВУЗ: 

КФУ | Казань

Составители: 

Рубрика: 

1) −2[a, b]; 2) [a, b]; 3)
3
4
[a, b] {6, −3, −3} {−12, −26, −8}
{0, 0, 0} 18
√
2 1) − 7; 2) {−46, 29, −12}; 3) {−7, 7, 7}
b
1
=
[a
2
,a
3
]
(a
1
,a
2
,a
3
)
b
2
=
[a
3
,a
1
]
(a
1
,a
2
,a
3
)
b
3
=
[a
1
,a
2
]
(a
1
,a
2
,a
3
)
x =
α[b,c]+β[c,a]+γ[a,b]
(a,b,c)
b
1
= {−
2
3
,
4
3
, −1} b
2
= {
1
3
,
1
3
, 1} b
3
= {
2
3
, −
1
3
, 1}
x + 2y + z −9 = 0 x + y −2 = 0 x −2y = 0 2x + z = 0
4y+z = 0 10x+9y+5z− 74 = 0 x = 2−5u+4v y = 3+6u−2v
z = −5 + 4u x = −13 y = 13 z = −9 u = −
1
5
v =
2
5
x−4y −z +16 = 0 x+5y −z +5 = 0 A B
D E
C F 6x + 9y − 22z = 0
5y + 13z − 60 = 0 10x − 7z = 0 6y − 7 = 0
39x−29y−7z = 0 5x−6y−7z+41 = 0 27x+11y+z−65 = 0
13x + y −20 = 0 x = −6 y = −4 z = −3 u + v −1 = 0
u = 0 v = 0 39u + 9v − 1 = 0 2x + 3y + 4z − 1 = 0
x + 3y + 9 = 0 z − 1 = 0 20x + 19y − 5z + 41 = 0
(3, 5, 7)
x = 2+2t
y = 3 + 3t z = 1 + 8t x = 7 − 2t y = −1 z = 2 + t x = 1
y = t z = 1 x = −2t y = 7t z = 4t x = t y = −8 − 4t
z = −3 − 3t A, B D C E
x − 3y + 5z = 0 (−3, 5, −5)
                                               ÎÒÂÅÒÛ


     1. 1) −2[a, b]; 2) [a, b]; 3) 34 [a, b]. 4. {6, −3, −3}, {−12, −26, −8},
                                √
{0, 0, 0}. 5. 18 2. 6. 1) − 7; 2) {−46, 29, −12}; 3) {−7, 7, 7}.
7. b1 = (a[a1 ,a2 ,a2 ,a3 ]3 ) , b2 = (a[a1 ,a3 ,a2 ,a1 ]3 ) , b3 = (a[a1 ,a1 ,a2 ,a2 ]3 ) . 8. 37,5. 9. 1) 25; 2)
0. 14. Ðàâåíñòâî èìååò ìåñòî òîãäà è òîëüêî òîãäà, êîãäà âûïîëíå-
íî, ïî êðàéíåé ìåðå, îäíî èç äâóõ óñëîâèé: 1) âåêòîð b ïåðïåíäèêó-
ëÿðåí ê âåêòîðàì a è c; 2) âåêòîðû a è c êîëëèíåàðíû. 15. x =
α[b,c]+β[c,a]+γ[a,b]
       (a,b,c)               . 16. b1 = {− 23 , 43 , −1}, b2 = { 13 , 31 , 1}, b3 = { 23 , − 13 , 1}.
17. 1) x + 2y + z − 9 = 0; 2) x + y − 2 = 0. 18. x − 2y = 0, 2x + z = 0,
4y+z = 0. 19. 10x+9y+5z−74 = 0. 20. x = 2−5u+4v , y = 3+6u−2v ,
z = −5 + 4u. 21. 1) x = −13, y = 13, z = −9; 2) u = − 51 , v = 25 .
22. 1) x−4y −z +16 = 0; 2) x+5y −z +5 = 0. 23. Òî÷êè A è B ëåæàò
â äàííîé ïëîñêîñòè, òî÷êè D è E  ïî îäíó ñòîðîíó îò ïëîñêîñòè, à
òî÷êè C è F  ïî äðóãóþ ñòîðîíó îò íåå.                                   24. 6x + 9y − 22z = 0.
25. 5y + 13z − 60 = 0. 26. 1) 10x − 7z = 0; 2) 6y − 7 = 0; 3)
39x−29y−7z = 0. 27. 5x−6y−7z+41 = 0. 28. 27x+11y+z−65 = 0.
29. 13x + y − 20 = 0. 30. 1) x = −6, y = −4, z = −3; 2) u + v − 1 = 0,
u = 0, v = 0; 3) 39u + 9v − 1 = 0. 31. 2x + 3y + 4z − 1 = 0,
x + 3y + 9 = 0, z − 1 = 0. 32. 20x + 19y − 5z + 41 = 0. 33. 1)
Òðè ïëîñêîñòè ïåðåñåêàþòñÿ â òî÷êå (3, 5, 7); 2) òðè ïëîñêîñòè ïîïàð-
íî ïàðàëëåëüíû; 3) òðè ïëîñêîñòè ïðîõîäÿò ÷åðåç îäíó ïðÿìóþ; 4)
ïëîñêîñòè ïîïàðíî ïåðåñåêàþòñÿ è ëèíèÿ ïåðåñå÷åíèÿ êàæäûõ äâóõ
ïëîñêîñòåé ïàðàëëåëüíà òðåòüåé ïëîñêîñòè; 5) ïåðâàÿ è òðåòüÿ ïëîñ-
êîñòè ïàðàëëåëüíû, âòîðàÿ ïëîñêîñòü èõ ïåðåñåêàåò. 34. 1) x = 2+2t,
y = 3 + 3t, z = 1 + 8t; 2) x = 7 − 2t, y = −1, z = 2 + t; 3) x = 1,
y = t, z = 1. 35. 1) x = −2t, y = 7t, z = 4t; 2) x = t, y = −8 − 4t,
z = −3 − 3t. 36. Òî÷êè A, B è D ëåæàò íà ïðÿìîé, òî÷êè C è E
íåò. 37. x − 3y + 5z = 0. 38. 1) ïåðåñåêàþòñÿ â òî÷êå (−3, 5, −5)

                                                       56

Страницы