, i, j, k . (a 1, a 2, a 3), (b 1, b 2, b 3). , .
= (a 1 i + a 2 j + a 3 k) (b 1 i + b 2 j + b 3 k) = a 1 b 1 ii + a 1 b 2 ij + a 1 b 3 ik +
+ a 2 b 1 ji + a 2 b 2 jj + a 2 b 3 jk + a 3 b 1 ki + a 3 b 2 kj + a 3 b 3 kk.
, i, j, k Þ ii = = jj = kk = 1, ij = ik = jk = 0, .
= a 1 b 1 + a 2 b 2 + a 3 b 3. (7)
Þ 2 = = a 12+ a 22+ a 32 (8)
Þ ½½= = (9)
Þ cosÐ(, ) = =. (10)
A (x 1, y 1, z 1), B (x 2, y 2, z 2), (x 2 x 1, y 2 y 1, z 2 z 1) Þ
½½=. (11)
r(A, B) A B. r(A, B) (11). , :
1. r(A, B) = r(B, A);
2. r(A, B) + r(B, C) ³ r(A, C) ( );
3. r(A, B) ³ 0, r(A, B) = 0 Û A = B.
. , , , , . 2 3 . , .
5. , : (a 1, a 2, a 3) (b 1, b 2, b 3), :
(12)
= (a 1 b 2 a 2 b 1) i (a 1 b 3 a 3 b 1) j + (a 2 b 3 a 3 b 2) k.
. , . , .
1.
½½2= (a 2 b 1 a 3 b 1)2 + (a 1 b 3 a 3 b 1)2 + (a 2 b 3 a 3 b 2)2 (*),
(½½½½sinÐ(, ))2 = ½½2½½2sin2Ð(, ) =½½2½½2(1 cos2Ð(, )) =
=½½2½½2 ½½2½½2cos2Ð(, ) =½½2½½2 ( )2 =
= (a 12+ a 22+ a 32) (b 12+ b 22+ b 32 ) (a 1 b 1 + a 2 b 2 + a 3 b 3)2. (**)
(*) (**), , .
2. = (a 1 i + a 2 j + a 3 k) ()
. ^ . - , ^.
3. ½½ , . . , Ox , Oy , , . Oz
|
|
. (a 1,0, 0), (b 1, b 2, 0), , a 1> 0, b 2> 0. (12) = a 1 b 2 k, , a 1 b 2> 0. Oz, , (, , ) .
, , .
1. ´ = ´.
, , :
2. (l )´ = l( ´ ).
, , :
3. ´( + ) = ´ + ´.
,
4. i ´ j = k, k ´ i = j, j ´ k = i.
(12).
. , .