Задачи по аналитической геометрии. Часть II. Игудесман К.Б. - 26 стр.

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(AB)
(P QR)
A(1, 1, 1, 1), B(−2, −1, 1, 3), P (2, 1, −1, 0), Q(3, 1, 0, −1), R(0, 0, −1, 1)
A(0, 0, 1, 1), B(2, −1, 0, 0), P (3, 1, 0, −2), Q(−2, 0, −1, 3), R(1, 1, 1, −1)
L
2x
1
+ x
2
+ 3x
3
− x
4
= 0
3x
1
+ 2x
2
− 2x
4
= 0
3x
1
+ x
2
+ 9x
3
− x
4
= 0 .
L
∗
y
z x = (5, 2, −2, 2) L
a
1
= (2, 1, 1, −1), a
2
= (1, 1, 3, 0), a
3
= (1, 2, 8, 1)
x = (1, 0, 3, 0)
L a
1
= (5, 3, 4, −3), a
2
=
(1, 1, 4, 5), a
3
= (2, −1, 1, 2)
M
1
M(0, −1, 2, 1) Π : 2x
1
+ x
2
+ x
4
− 3 = 0
A(2, 3, −1, 4)
Π : x
1
− x
2
+ 2x
3
+ x
4
+ 1 = 0
M
1
M(2, −1, 3, 1)
(
x
1
+ x
2
+ x
3
− x
4
+ 1 = 0
2x
1
+ x
2
− 2x
3
+ x
4
+ 2 = 0 .
A(3, 1, −1, 1) l
x
1
= −λ, x
2
= λ + 2, x
3
= −λ + 1, x
4
= 2λ .
A Π
2
   104. Íàïèñàòü óðàâíåíèÿ îáùåãî ïåðïåíäèêóëÿðà ïðÿìîé (AB) è
ïëîñêîñòè (P QR):
1) A(1, 1, 1, 1), B(−2, −1, 1, 3), P (2, 1, −1, 0), Q(3, 1, 0, −1), R(0, 0, −1, 1);
2) A(0, 0, 1, 1), B(2, −1, 0, 0), P (3, 1, 0, −2), Q(−2, 0, −1, 3), R(1, 1, 1, −1).
                          
   105. Ëèíåéíîå ïîäïðîñòðàíñòâî L çàäàíî óðàâíåíèÿìè:
                       
                          1   2    3   4
                        2x + x + 3x − x = 0
                       
                               3x1 + 2x2 − 2x4 = 0
                       
                       
                        3x1 + x2 + 9x3 − x4 = 0 .

Íàéòè óðàâíåíèÿ, çàäàþùèå îðòîãîíàëüíîå äîïîëíåíèå L∗ .
   106. Íàéòè îðòîãîíàëüíóþ ïðîåêöèþ y è îðòîãîíàëüíóþ ñîñòàâëÿ-
þùóþ z âåêòîðà x = (5, 2, −2, 2) íà ëèíåéíîå ïîäïðîñòðàíñòâî L, íà-
òÿíóòîå íà âåêòîðû a1 = (2, 1, 1, −1), a2 = (1, 1, 3, 0), a3 = (1, 2, 8, 1).
   107.    Íàéòè óãîë ìåæäó âåêòîðîì x = (1, 0, 3, 0) è ëèíåéíûì
ïîäïðîñòðàíñòâîì L, íàòÿíóòûì íà âåêòîðû a1 = (5, 3, 4, −3), a2 =
(1, 1, 4, 5), a3 = (2, −1, 1, 2).
   108. Âû÷èñëèòü êîîðäèíàòû îðòîãîíàëüíîé ïðîåêöèè M1 òî÷êè
M (0, −1, 2, 1) íà ãèïåðïëîñêîñòü Π : 2x1 + x2 + x4 − 3 = 0.
   109. Âû÷èñëèòü ðàññòîÿíèå îò òî÷êè A(2, 3, −1, 4) äî ãèïåðïëîñ-
êîñòè Π : x1 − x2 + 2x3 + x4 + 1 = 0.
   110. Âû÷èñëèòü êîîðäèíàòû îðòîãîíàëüíîé ïðîåêöèè M1 òî÷êè
M (2, −1, 3, 1) íà ïëîñêîñòü
                    (
                        x1 + x2 + x3 − x4 + 1 = 0
                        2x1 + x2 − 2x3 + x4 + 2 = 0 .
   111. Âû÷èñëèòü ðàññòîÿíèå îò òî÷êè A(3, 1, −1, 1) äî ïðÿìîé l:

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

   112. Âû÷èñëèòü ðàññòîÿíèå îò òî÷êè A äî ïëîñêîñòè Π2 :


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