Векторная алгебра

1 x , a = (2;1;−1) ,.

, (x, a ) = 3 .

31

'

a

b * *, * "

r

= 5,

b

= 12 .

a

r

r

r

' "

+ b

− b

.

a

a

32

1.11. (! ! !

. &$ (!$ a b

! c , ,. ,. :

r

r

r

r r

1)

c

=

a

b

sin a,b ,

r

2)

c a ,

b ,

c

3)

r

r r

- ! ! !.

a;b;c

' * "

r

× b

a

* , "

r

× b

a

*

, * ( a b

, , . 2.22.

r r	] .	1
[a;b

" * .

r

( a |/| b ). ) 1-3

' * (

. ' ",

r r	] = 0 ,	r r	r
[a; a		[a;0]	= 0 .

' * ,. :

r r			r r	] ,
1) [a;b	]= −[b; a
r	r		r	r	r	r
2) λ[a;b		]= [λa;b ]= [a;λb ] , λ - # ! + " #.
r		r	r		r r
3) [a;b	+ c		]= [a;b ]+ [a; c ] .
		(! ! 2$
$ . 2 a b
					r	= (ax ;a y ; az ),	b = (bx ; by ; bz )
					a			v
					i; j;k		r		]
							, * [a;b
*			"

33

r r

i

j

k

[a, b ]=

ax

a y

az

.

(20)

bx

by

bz

/ *, * "

r

r

]=

a

a

i −

a

a

j +

a

a

r

[a, b

y

z

x

z

x

y

k .

by

bz

bx

bz

bx

by

r

=

r

r

− 3k ,

r

- $ . ! a

2i +

j

b = i + 2k .

r

r

r

+ b, b ]

* . *

,

*

1 [2a

( a b .

r

r

r

+ b

, b ]

* .

/. ' [2a

1. * 3 2 *, * "

r

r

[b, b]=

r

[2a + b, b]=

2[a, b]+

2[a, b] .

r

) "

(20)

[r

r]

=

i

j

k

r .

a

, b

2 1 − 3

= 2i − 7 j − k

1

0

2

r

+ b, b]

= 2(2i − 7 j − k )= 4i − 14 j − 2k .

" [2a

2. !

r

+ b

= 2(2;1; −3) + (1; 0; 2) = (5; 2; −4), " (20)

2a

[ r

]

=

i

j

k

=

2 − 4

i −

5 − 4

j +

5 2

k

= 4i − 14 j − 2k

.

2a

+ b, b

5 2 − 4

0

2

1

2

1

0

1

0

2

r

, b]

=

2i − 7 j − k

=

! . *

54 .

[a

r

S =

54 .

!: [2a + b, b]= 4i − 14 j − 2k ,

1.12. / !&. ( "

3 ". %

r

= 10,

b

r

a

= 2 (a, b )= 12 .

' ":

r

r

.

[a, b ]

34

r

r

r

12

3

r

=

a

b

=

=

/. cos a,b

r

r

,

a

b

20

5

r r

3 2

sin a,b

=

1 −

5

r

r

r

r

r r

[a, b ]

=

a

b

sin a,b

r

r

!:

= 16 .

[a;b ]

r

= 3i

− j − 2k ,

b = i + 2

3 ". %: a

r r

r

r

r

r

1: ) [a, b ] ,

) [3a

− 2b; a + 2b ] .

= 0,8 ,

= 10 2 0,8 = 16 .

j − k .

r

]=

i

j

k

r

3

− 1

− 2

= 5i + j + 7k .

/: ) [a

, b

1

2

− 1

r

−

r

r

r

r

r

]− 4[b, b] =

) [3a

2b; a + 2b]= 3[a, a ]+ 6[a, b]− 2[b, a

r

r

r

r

r

r

] = 8(5i + j + 7k )= 40i + 8 j + 56k .

= 6[a, b

]+ 2[a

, b

]= 8[a

, b

!: )

r r

r

r

r

[a;b ]

= 5i +

j

+ 7k ;

r

r

+

r

r

r

) [3a

− 2b; a

2b] = 40i

+ 8 j +

56k .

3 ". ' " * . - " (

A(− 1;−2;0), B(3; 2;1), C (− 1;−4; 2).

	/.	AB = (4; 4;1), AC = (0;−2; 2).
				1		[AB, AC ]				1		i	j	k		1
	S	ABC	=						=			4 4		1	=			10i − 8 j − 8k	=	57	.
			2						2						2
												0	− 2	2
	!: S ABC =
					57 .

35

1.13. 3 " #$# +

&

' a b * *.

0,

"

r

= 3,

r

= 4 ,

b

a

":

r r

r

r

r

)

r

,

)

r

r

.

[a

+ b , a

− b ]

[3a − b , a − 2b ]

36

% " A(1;2;0), B(3;0;−3) C (5;2;6). ' " * .

&').

37

1 x , , " * * a = (2; −3;1)

b = (1; −2; 3) ,	r	(i + 2 j − 7k )= 10 .
	x

38

' * R 3 ": A(1;1; 2), B(3;1; −1), C (0; −2; 2), D(3; 4; −2). 1 " M (x; y; z ), , " DM AB DM AC DM = 18 .

39

1.14. $/ (! ! !

.	$/ &$ (!$ (						r	r
							a	, b , c
	r	r	c .
* [a, b ]
"	r	r	r	r	r	. ",
	- * ([a	, b	] , c )	a	b c

" - * * " ". ) -

* ,. .

r

r

" # #!

# !

1.

r

r

r

r

]).

([a

, b ] , c )= (a

,[b , c

2.

r

r

= b

r

r

=

r

r

b .

a

b c

c

a

c

a

3.

r

r

r

r

r

r

a

b c

= −b a c

= −a

c b .

$" # #!

# !

4.

-

*

"

+

* * *,

*

*

( (:

V =

r

r

r

.

a

b c

+

*

VABCD , * (

r

r

( .

a, b , c

2.25), 16 + * * *:

1

r

r

r

VABCD

=

a

b

c

.

6

5. $/ (! ! % ) ,

r

r

$&.

! & a, b , c

r r

6.

2

r

r

r

-

*;

r

r

<

a

b c > 0 ,

a

, b , c

a

b c

r r r

.

a, b, c

$/ (! !

2$

r

r

$ . 2 a

, b , c

r

, a y , az ),

b = (bx , by , bz ),

r

, cz )

a = (ax

c = (cx cy

,

0 ,

40