Задачи по аналитической геометрии. Часть II. Игудесман К.Б. - 23 стр.

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m
T
(AB) m
T
Π
2
A(−1, 0, 2, 3) B(2, 1, −1, 0)
Π
2
:
(
x
1
+ x
2
− x
3
= 0
2x
2
+ 2x
3
− x
4
+ 3 = 0 .
a
1
= {1, 1, 1, 1, 0}, a
2
= {1, 1, −1, −1, −1}, a
3
= {2, 2, 0, 0, −1}, a
4
=
{1, 1, 5, 5, 2}, a
5
= {1, −1, −1, 0, 0}
a
1
= {1, −1, 1, −1, 1}, a
2
= {1, 1, 0, 0, 3}, a
3
= {3, 1, 1, −1, 7}, a
4
=
{0, 2, −1, 1, 2}
a
1
= {1, 2, 1, −2}, a
2
= {2, 3, 1, 0}, a
3
= {1, 2, 2, −3}
b
1
= {1, 1, 1, 1}, b
2
= {1, 0, 1, −1}, b
3
= {1, 3, 0, −4}
R
n
L
1
x
1
+ x
2
+ . . . + x
n
= 0
L
2
x
1
= x
2
= . . . = x
n
a
0
+ a
1
t b
0
+ b
1
t
a
0
= {3, 1, 2, 1, 3}, a
1
= {1, 0, 1, 1, 2}
b
0
= {2, 2, −1, −1, −2}, b
1
= {2, 1, 0, 1, 1}
A(0, −3, 0, 2), B(1, 1, −1, −2), C(−1, −7, 1, 6)
(ABC) (A
1
B
1
C
1
)
A(1, 2, 0, −1) B(2, 1, −1, 0) C(0, 3, −4, 1)
A
1
(2, 0, 1, −3) B
1
(2, 0, 11, −9) C
1
(3, −1, −5, 1)
Π [AM)
Π : 3x
1
+ 2x
2
+ x
3
− 2x
4
+ 4 = 0
          T              T
òî÷åê m       (AB) è m      Π2 , åñëè A(−1, 0, 2, 3), B(2, 1, −1, 0),
                            (
                                     x1 + x2 − x3 = 0
                     Π2 :
                              2x2 + 2x3 − x4 + 3 = 0 .
                            
   88. Íàéòè ðàçìåðíîñòü è áàçèñ ëèíåéíîãî ïîäïðîñòðàíñòâà, íàòÿ-
íóòîãî íà ñëåäóþùóþ ñèñòåìó âåêòîðîâ:
a1 = {1, 1, 1, 1, 0}, a2 = {1, 1, −1, −1, −1}, a3 = {2, 2, 0, 0, −1}, a4 =
{1, 1, 5, 5, 2}, a5 = {1, −1, −1, 0, 0}.
   89. Íàéòè ñèñòåìó ëèíåéíûõ óðàâíåíèé, çàäàþùóþ ëèíåéíîå ïîä-
ïðîñòðàíñòâî, íàòÿíóòîå íà ñëåäóþùóþ ñèñòåìó âåêòîðîâ:
a1 = {1, −1, 1, −1, 1}, a2 = {1, 1, 0, 0, 3}, a3 = {3, 1, 1, −1, 7}, a4 =
{0, 2, −1, 1, 2}.
   90. Íàéòè áàçèñ ñóììû è ïåðåñå÷åíèÿ ëèíåéíûõ ïîäïðîñòðàíñòâ,
íàòÿíóòûõ íà ñèñòåìû âåêòîðîâ
a1 = {1, 2, 1, −2}, a2 = {2, 3, 1, 0}, a3 = {1, 2, 2, −3};
b1 = {1, 1, 1, 1}, b2 = {1, 0, 1, −1}, b3 = {1, 3, 0, −4}.
   91. Äîêàçàòü, ÷òî ïðîñòðàíñòâî Rn åñòü ïðÿìàÿ ñóììà äâóõ ëèíåé-
íûõ ïîäïðîñòðàíñòâ: L1 , çàäàííîãî óðàâíåíèåì x1 + x2 + . . . + xn = 0,
è L2 , çàäàííîãî ñèñòåìîé óðàâíåíèé x1 = x2 = . . . = xn .
   92. Íàéòè òî÷êó ïåðåñå÷åíèÿ äâóõ ïðÿìûõ a0 + a1 t è b0 + b1 t:
a0 = {3, 1, 2, 1, 3}, a1 = {1, 0, 1, 1, 2};
b0 = {2, 2, −1, −1, −2}, b1 = {2, 1, 0, 1, 1}.
   93. Äîêàçàòü, ÷òî òî÷êè A(0, −3, 0, 2), B(1, 1, −1, −2), C(−1, −7, 1, 6)
ïðèíàäëåæàò îäíîé ïðÿìîé.
   94. Äîêàçàòü, ÷òî ïëîñêîñòü (ABC) ïàðàëëåëüíà ïëîñêîñòè (A1 B1 C1 ):
A(1, 2, 0, −1), B(2, 1, −1, 0), C(0, 3, −4, 1);
A1 (2, 0, 1, −3), B1 (2, 0, 11, −9), C1 (3, −1, −5, 1).
   95. Íàéòè ïåðåñå÷åíèå ãèïåðïëîñêîñòè Π è ëó÷à [AM ):
1) Π : 3x1 + 2x2 + x3 − 2x4 + 4 = 0,

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