Задачи по аналитической геометрии. Часть II. Игудесман К.Б. - 5 стр.

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(a, b, c) = 0
(a, b, c) = a[b, c] = [a, b]c
(a, b, c) = (c, a, b) = (b, c, a) = −(b, a, c) = −(c, b, a) = −(a, c, b)
a = {X, Y, Z} b = {X
0
, Y
0
, Z
0
}
c = {X
00
, Y
00
, Z
00
}
(a, b, c) =
¯
¯
¯
¯
¯
¯
¯
¯
X Y Y
X
0
Y
0
Z
0
X
00
Y
00
Z
00
¯
¯
¯
¯
¯
¯
¯
¯
.
1) [(a + b), (a − b)]; 2) [a, (a + b)]; 3)
·
a + b
2
,
³
b −
a
2
´
¸
.
[a, b] = [b, c] = [c, a] a + b + c = 0
[a, b] + [b, c] + [c, a]
[a, b]
1) a = {2, 3, 1}, b = {5, 6, 4};
2) a = {5, −2, 1}, b = {4, 0, 6};
3) a = {−2, 6, −4}, b = {3, −9, 6}.
a = {8, 4, 1}, b = {2, −2, 1}
áàçèñîì i, j, k , è îòðèöàòåëüíîå â ïðîòèâíîì ñëó÷àå. Åñëè âåêòîðû a,
b, c êîìïëàíàðíû, òî (a, b, c) = 0 ïî îïðåäåëåíèþ.
   Ñâîéñòâà ñìåøàííîãî ïðîèçâåäåíèÿ:

  1. (a, b, c) = a[b, c] = [a, b]c.

  2. (a, b, c) = (c, a, b) = (b, c, a) = −(b, a, c) = −(c, b, a) = −(a, c, b).

   Åñëè â îðòîíîðìèðîâàííîì áàçèñå a = {X, Y, Z}, b = {X 0 , Y 0 , Z 0 },
c = {X 00 , Y 00 , Z 00 }, òî
                                             ¯              ¯
                                             ¯ X Y Y ¯
                                             ¯              ¯
                                             ¯              ¯
                                (a, b, c) = ¯¯ X 0 Y 0 Z 0 ¯¯ .
                                             ¯ 00           ¯
                                             ¯ X Y 00 Z 00 ¯

                                         ÇÀÄÀ×È
   1. Çíàÿ äâà âåêòîðà a è b, íàéòè:
                                                             ·             ¸
                                                                 a+b ³   a´
     1) [(a + b), (a − b)];          2) [a, (a + b)];   3)          , b−     .
                                                                  2      2
   2. Ïîêàçàòü, ÷òî åñëè òðè âåêòîðà a, b, c íå êîëëèíåàðíû, òî èç
ðàâåíñòâ [a, b] = [b, c] = [c, a] âûòåêàåò ñîîòíîøåíèå a + b + c = 0, è
îáðàòíî.
   3. Èç îäíîé òî÷êè ïðîâåäåíû òðè íåêîìïëàíàðíûõ âåêòîðà a, b,
c. Ïîêàçàòü, ÷òî ïëîñêîñòü, ïðîõîäÿùàÿ ÷åðåç êîíöû ýòèõ âåêòîðîâ,
ïåðïåíäèêóëÿðíà ê âåêòîðó [a, b] + [b, c] + [c, a].
   4. Íàéòè âåêòîðíîå ïðîèçâåäåíèå [a, b] â êàæäîì èç íèæåñëåäóþ-
ùèõ ñëó÷àåâ:

                        1) a = {2, 3, 1},          b = {5, 6, 4};
                        2) a = {5, −2, 1},         b = {4, 0, 6};
                        3) a = {−2, 6, −4}, b = {3, −9, 6}.
   5. Âû÷èñëèòü ïëîùàäü ïàðàëëåëîãðàììà, ïîñòðîåííîãî íà âåêòî-
ðàõ a = {8, 4, 1}, b = {2, −2, 1}.

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