Векторная алгебра

21

' * "D	r	,
	AC = a
* . ' " a

CD , DA , * ,. * .

BD = b , * ,.

b AB , BC ,

22

* " a = (2;−3;−1), *

" B(1;−1;2). 1 a0 a .

23

% ax = 4 ,					a y = −12 a . * ,
a z * , "		r			= 13.
		a

24

1 *				p = (2;−3)	q = (9;4). !,
,			,	,,
d = (49;14) *		r r
		p, q .

25

1.8. (! ! !

. &$ (!$ ( a b

" # ( ), * ,

/ (

r

, b )

. " * (a

r

a b :

r

r

r

r

r r

(a, b )=

a

b

a,b

(14)

cos

.

) *

* .

r

r

r

r

(a, b )=

a

!a b =

b

!ba .

(15)

%,

r

r

r

r

r

r r

r

b

cos(a, b )= ! b ;

a

cos

= ! a .

a

a,b

b

(15) , " " ( *

r

r

!a b =

(a, b )

r

(a,b )

r

;

!ba =

r

.

(16)

a

b

) * ,. .

	r		r	" # #! # !
1. (a, b )= (b , a ).
2.		r	r
	(λa, b )= λ (a, b ), λ - ".
3.	r	r	r	r r
	(a	, b + c )= (a		, b )+ (a, c ).

(r r) = r 2

4.a, a a .

3$". ) *

r

r

(a, a ) # &$

r

2 , . .

!$ a " a

r r

r

r

2 .

(a, a ) = a 2 =

a

$" # #!

# !

r

5. 2 a b * *, (a, b )= 0 .

r r

r

r

cosπ

2 =

r

r

0 = 0 .

%, (a, b )=

a

b

a

b

r

) * , ( , . .

= 0 b = 0 .

a

6. 2 a b ,

26

r			r r
(a, b )> 0	, 0	<	a,b <			π 2 ,
r				r r
(a, b )< 0 , π		2 <		a,b < π .

& " * , * (

* ( ( ( .

r

r

r

b

r

= 2 ,

r

= 3 .

b

- $ . ' " (2a − b , a + 4b ), a

a

/. * * 3, * "

r

r

r

r

r

r

(2a

− b , a +

4b )= (2a

, a ) + (− b , a )+

(2a,4b )+ (− b , 4b )=

=

2

r

r

r

= 2a 2

− (b, a )+ 8(a, b )− 4b 2 =

=

1

r

+

r

r

r

= 2a 2

7(a, b )

− 4b 2 =

=

r

r

r

r

( 4)

=

(a, b )= 0, a b

r

− 4b 2 = 2 4 − 4 9 = −28 .

= −2a 2

!: − 28 .

(! !

2$

0 ! '. # ! & a b ( & #!$ $ :

r

b = (bx , by , bz )

a = (ax , a y , az );

! $!$ ( #

r

r

r

, # (!

i , j , k

r

, b ) # 2$%

(a

r

+ a y by

+ azbz .

(a, b )= axbx

(17)

( +# !. !

, " (i,i )=

i

2 = 1 . & "

r

r

% (

" (

(j , j )= (k , k )= 1 .

i, j, k

* *

(i, j )= (i, k )= (j, k )= 0 ,

* / " 1-3, * "

r

+ az k , bx i + by j + bz k )= axbx (i,i )+ axby ×

(a, b )= (ax i + a y j

×(i, j )+ ax bz (i, k )+ a y bx (j,i )+ a y by (j, j )+

+a y bz (j, k )+ azbz (k ,i )+ azby (k , j )+ azbz (k , k )=

27

= a x bx 1 + a x by 0 + a x bz 0 + a y bx 0 + a y by 1 + a y bz 0 +

+ az bx 0 + az by 0 + az bz 1 = axbx + a y by + az bz . ■

3$". 1. (17) * *

		r
* * ( ((a, b )= 0):
r	b	axbx + a y by + azbz = 0 .
a

2.(14) " (17) (

,

r

r

(r

)

a b + a

y

b

y

+ a

z

b

os(a

, b )=

a

, br

=

x

x

z

.

(18)

r

ax2 + a 2y + az2 :

a

b

bx2 + by2 + bz2

0

, "

r

r

r

a

= (a, a ) .

3.

'

(16)

" (

* * * ,

!a b =

ax b

x

+ a y by

+ az

bz

, !b b =

ax bx + a y by

+ az

bz

.

(19)

a 2

+ a2

b2 + b2

+ a2

+ b2

x

y

z

x

y

z

- $ . 1

r

= 3i − 4 j

b = 6i + 2 j − 3k ,

a

!r b .

a

r

/. !

=

(3;−4;0);

b

= (6;2;−3),

a

(17)

r r

3 6 + (− 4) 2 + 0 (− 3)

10

2

os a,b =

=

=

;

2

2

0

2

6

2

+

2

2

2

3

+ (− 4) +

+ (− 3)

5 7 7

r r

2

3 6 + (− 4) 2 + 0 (− 3)

10

= arccos

,

!r b =

=

= 2 .

a,b

a

2

2

7

3

+ (− 4)

5

'

(

*

(

*

*

" A F * * .

" M

" *

. S ( . 2.20):

A = (F , S ).

28

1.9. / !&. ( "

3 ". ' a b , ϕ

=

2π

. 0, "

r

=

3 ,

r

= 4 ,

a

b

3

r

− 2b ,

r

" (3a

a + 2b ).

/

r

r

r r

r

r

(3a − 2b ,

a + 2b )= 3(a; a ) + 6(a;b )− 2(b; a )− 4(b;b )=

r

2

r

2

r r

2π

=

3

a

−

4

b

+ 4(a;b )= 3 9 − 4 16 + 4

a

b

cos

=

3

= 27 − 64 + 4 3 4 (− 1 2) = −61.

!: -61.

r

r

3 ". %:

= 3 ,

= 5 . *, *

" α

b

a

r

+ αb

r

− αb

* *.

a

, a

/. ! *

r

r

, " (a + αb ) (a − αb ),

r

r

− αb )=

r

r

r

r

(b, b )= 0

(a

+ αb , a

0 (a, a ) − α (a, b )+ α (b

, a )− α 2

r

2

− α

2

= 9

9 − 25α

2 = 0, α 2 =

9

α

= ±

3

a

b

.

3

25

5

!: α = ±

.

5

A(3;2;4),

B(5;1;−1),

C (1;−2;1).

3 ". %

-

* - * - A ( . 2.21).

/.

AB = (5 − 3; 1 − 2; − 1 − 4) = (2; − 1; − 5);
AC = (1 − 3; − 2 − 2; 1 − 4) = (− 2; − 4; − 3);
cos AB, AC							= AB AC = 2(− 2) + (− 1)(− 4) + (− 5) (− 3) =
				^
										AB				AC			4 + 1 + 25 4 + 16 + 9
=		15				=			15		.
		30		29			58
!: - * - A 2π − arccos																							15	.
																							58
															29

1.10. 3 " #$# + &

% a = (4; −2; −4),				b = (6; −3; 2). ' ":
r			r			r
					) b 2 ,
) (a, b ),			) a 2 ,
r	r	+ 2b),	r	r 2	r		r 2
) (2a	− 3b; a		) (a	+ b ) ,	) (a	− b ) .

30