,
, . , .
2>1>0, 3<0.
3, 1, 2, - Q, F, N, M . :
Q (3), F (x1), N(x2), M (x).
F N FN. , | FN | = 2- 1.
. - , - .
, .
x, (, ).
. , z. , , z. :
(, , z).
. .
.
, , ( ).
:
| | ,
| | .
( -), . , 0. : | | = 0.
, . , : || .
, , . : = .
, , .. .
, .
, .
, .
, , , , , .
, , : = = .
MNKZ , , , , , . , , = = .
, ABCD.
= , ¹ , ¹ , , , :
| | = | | = | | = | |.
.
+ , , . :
|
|
= + .
. 1. . 2.
" " (. 1). " " (. 2): , , , .
, " ": , , . . 3 + + + .
. 3
" ": , , , , , , , (. 4):
= + + .
. 4
× , , , | |×| |, >0 <0. :
= × .
=0, × -:
0 × = .
=1, 1× = .
= -1, (-1)× =- - , .
, , . , , = × , - , . , = × .
, : = , = .
, .
, 2:1, . =2/3× , D .
=1/2× =1/2× ; =-1/2× .
D = + = -1/2× + .
=-2/3(-1/2 + )= 1/3× -2/3× .
, =1/3× -2/3× . , , :
= +(1)× = +( ).
D = =1/2× .
1/| |, ( ), 0. ,
0=1/| |× = /| |; | 0|=1.
, , z , , .
, . :
1) + = + ;
2) +( + )=( + )+ ;
3) ×( × ) = ( × )× ;
|
|
4) ×( + )= × + × ;
5) ( + )× = × + × .
, , . .
.
. 5
. 6
, .
: =(, ) (. 5).
, z.
: =(, , z) (. 6).
, , .
=(1, 1, z1), =(2, 2, z2)
= + ; = - ; = × ,
, , :
=(1+2; 1+2; z1+z2),
=(x1-x2; y1y2; z1z2),
=( ×1; ×1; ×z1).
. 5 . 6 , :
| |=| |= .
z .
, : . (1, 1, z1), . (2, 2, z2), .
. 7
. 7 , , . :
= ,
=(1, 1, z1), =(2, 2, z2).
:
=(21; 21; z2z1).
:
| |=| |= .
, .
()= .
, . . = .
, cos = = .
, , , , z .
cos = , cos = , cos = .
. :
cos2 +cos2 +cos2 =1.
, ,
= .
.
, .
× (, ).
,
× = × ×cos ,
.
:
1. × = ×
2.
3. ( + )× = × + ×
4. , ( ), . . ^ × =0.
× =0 .
5. × = . :
=
, , , . . =(1, 1, z1) =(2, 2, z2),
× =1×2+1×2+z1×z2
:
^ x1×x2+y1×y2+z1×z2=0
, , = (2, 1, 2), = (1, 0, 4), = (3, 4, 1).
|
|
× = 2 × 1 + (1) × 0 + 2 × 4 = 10,
× =2 × 3 + (1) × 4 + 2 × (1) = 0,
× = 1 × 3 + 0 × 4 + 4 × (1) = 1.
, .
,
.