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, 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).

. , .





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