Задачи по аналитической геометрии. Часть II. Игудесман К.Б. - 25 стр.

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a
1
, a
2
, . . . , a
p
E
n
g(a
1
, . . . , a
p
) =
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
(a
1
, a
1
) (a
1
, a
2
) . . . (a
1
, a
p
)
(a
2
, a
1
) (a
2
, a
2
) . . . (a
2
, a
p
)
. . . . . . . . . . . .
(a
p
, a
1
) (a
p
, a
2
) . . . (a
p
, a
p
)
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
.
a
1
, a
2
, . . . , a
p
p
L
∗
L a
1
= (1, 0, 2, 1), a
2
= (2, 1, 2, 3), a
3
=
(0, 1, −2, 1)
y
z x = (4, −1, −3, 4) L
a
1
= (1, 1, 1, 1), a
2
= (1, 2, 2, −1), a
3
= (1, 0, 0, 3)
x = (2, 2, 1, 1)
L a
1
= (3, 4, −4, −1), a
2
=
(0, 1, −1, 2)
M
1
M(1, 1, 1, −1) Π : x
1
+ x
2
− 2x
3
+ x
4
− 1 = 0
A(1, −1, 2, 1)
Π : x
1
+ 3x
2
− x
3
− x
4
+ 2 = 0
M
1
M(1, −2, 3, −1)
x
1
= λ − 2, x
2
= −λ + 2, x
3
= 2λ + 1, x
4
= −3λ .
A(1, 1, −2, 1) l
x
1
= λ, x
2
= −λ + 1, x
3
= λ + 2, x
4
= 2λ − 1 .
   Îïðåäåëèòåëåì Ãðàìà âåêòîðîâ a1 , a2 , . . . , ap åâêëèäîâà ïðîñòðàí-
ñòâà En íàçûâàåòñÿ îïðåäåëèòåëü
                                  ¯                                           ¯
                                  ¯                                           ¯
                                  ¯ (a1 , a1 ) (a1 , a2 )   ...   (a1 , ap ) ¯
                                  ¯                                           ¯
                                  ¯ (a , a ) (a , a )       ...   (a2 , ap ) ¯¯
                                  ¯ 2 1          2 2
            g(a1 , . . . , ap ) = ¯                                           ¯ .
                                  ¯ ...          ...        ...     . . . ¯¯
                                  ¯
                                  ¯                                           ¯
                                  ¯ (ap , a1 ) (ap , a2 )   ...   (ap , ap ) ¯

Îïðåäåëèòåëü Ãðàìà âåêòîðîâ a1 , a2 , . . . , ap ðàâåí êâàäðàòó p-ìåðíîãî
îáúåìà ïàðàëëåïèïåäà, ïîñòðîåííîãî íà ýòèõ âåêòîðàõ.

                                      ÇÀÄÀ×È
   97. Íàéòè áàçèñ îðòîãîíàëüíîãî äîïîëíåíèÿ L∗ ïîäïðîñòðàíñòâà
L, íàòÿíóòîãî íà âåêòîðû a1 = (1, 0, 2, 1), a2 = (2, 1, 2, 3), a3 =
(0, 1, −2, 1).
   98. Íàéòè îðòîãîíàëüíóþ ïðîåêöèþ y è îðòîãîíàëüíóþ ñîñòàâëÿ-
þùóþ z âåêòîðà x = (4, −1, −3, 4) íà ëèíåéíîå ïîäïðîñòðàíñòâî L, íà-
òÿíóòîå íà âåêòîðû a1 = (1, 1, 1, 1), a2 = (1, 2, 2, −1), a3 = (1, 0, 0, 3).
   99. Íàéòè óãîë ìåæäó âåêòîðîì x = (2, 2, 1, 1) è ëèíåéíûì ïîä-
ïðîñòðàíñòâîì L, íàòÿíóòûì íà âåêòîðû a1 = (3, 4, −4, −1), a2 =
(0, 1, −1, 2).
   100. Âû÷èñëèòü êîîðäèíàòû îðòîãîíàëüíîé ïðîåêöèè M1 òî÷êè
M (1, 1, 1, −1) íà ãèïåðïëîñêîñòü Π : x1 + x2 − 2x3 + x4 − 1 = 0.
   101. Âû÷èñëèòü ðàññòîÿíèå îò òî÷êè A(1, −1, 2, 1) äî ãèïåðïëîñ-
êîñòè Π : x1 + 3x2 − x3 − x4 + 2 = 0.
   102. Âû÷èñëèòü êîîðäèíàòû îðòîãîíàëüíîé ïðîåêöèè M1 òî÷êè
M (1, −2, 3, −1) íà ïðÿìóþ

           x1 = λ − 2, x2 = −λ + 2, x3 = 2λ + 1, x4 = −3λ .

   103. Âû÷èñëèòü ðàññòîÿíèå îò òî÷êè A(1, 1, −2, 1) äî ïðÿìîé l:

             x1 = λ, x2 = −λ + 1, x3 = λ + 2, x4 = 2λ − 1 .

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