Векторная алгебра

1.4. ( # # # !

1. * " e1, e2 ,..., en

* V ( #$ / *,

1);

2)a V .

a = λ1e1 + λ2e2 + ... + λn en .

(2)

, ,. .

$ 1. 6, % ! e V1 V1 .

$ 2. 6, ! &. e1 , e2 * V2

, V2 .

$ 3. 6, $&. e1, e2 , e3 *

V3 , V3 .

$ 4. ( '	(2) ) ! a			( #%	r	r	r
					e1	, e2	,..., en
# # ! V # !.
( +# !. ! *		*. ! . .
r	r	r	v
a =	α1e1 + α2e2 +		... + αnen .				(3)
' " (2) * " (3), * "
r		r	+ ... + (λn −	r
(λ1 − α1 )e1	+ (λ2 − α2 )e2			αn )en = 0 .			(4)
(4) * * . λi		− αi ≠ 0 , ( i = 1,2,..., n ,
r r	r			, / * "
", " e1, e2 ,..., en
* , * λ1 = αi		( i = 1,2,..., n .

$ 5 ( " # ! ! ( #). )

# # ! V " # ! ! ! !%. (! + &. ( # .

!.

2. # ! ! ( # *

, / *.

- $ . * * V1 ,V2 ,V3 .

/. 1,2,3 , " * V1 ,V2 ,V3

1,2,3.

11

1.5. & ! ! ( #

1. 122 , & (λ1 ; λ2 ;...; λn ) ( ' (2) !

a Vn

r

r

r

( #% e1

, e2 ,..., en

(&! ) # $ ! a ! 1$

( #.

r

V3 )

' ", * n = 3 (a

r

r

r

+

r

a = λ1e1 + λ2e2

λ3e3 ,

(5)

r

r

r

− V3 ,

(λ1;λ2 ;λ3 ) − a

e1

, e2 , e3

/

.

1

* ,

(5)

a

*

* , *:

r

(λ1;λ2 ;λ3 ).

4

a =

r

= (α1;α2 ;α3 ),

b = (β1;

β2 ; β3 ),

, " a

,

,

α1 = β1 , α2 = β2 , α3 = β3 .

6 *

,

, * , * ,.

*

:

1)

*

a

" α

, / ":

r

r

r

r

r

r

r

=

αa =

α (λ1e1 + λ2e2 + λ3e3 ) = αλ1e1 + αλ2e2 +

αλ3e3

=(αλ1 ;αλ2 ;αλ3 );

2)* ( ,.

r

r

r

r

r

r

r

! a

= λ1e1

+ λ2e2 +

λ3e3 ,

b = γ 1e1 + γ

2e2 + γ

3e3 .

3

r

r

r

r

+ (λ3 +

a

+ b = (λ1 + γ 1 )e1

+ (λ2 + γ 2 )e2

γ 3 )e3

r

+ b = (λ1 + γ 1;λ2 + γ 2 ,;λ3 + γ 3 ).

a

r

r

r

r

r

r

- $ . % ( a

= 2e1

− 3e2 + e3 ,

b = e1

+ 4e2

:

r

) 3a ,

+ 5b .

) − a

/:

r

r

r

r

r

r

r

−

) 3a = 3(2e1

3e2

+ e3 )

= 6e1

− 9e2

+ 3e3 = (6; −9; 3);

r

=

(2;−3;1);

b = (1;4;0)

r

r

r

) a

e1 , e2

, e3

,

r

+ 5b = (− 1)(2;−3; 1) + 5(1; 4; 0) = (3; 23; − 1).

− a

!: ) (6; −9; 3); ) (3; 23; − 1).

12

r	r	γ	r	,
' a = λ1e1 + λ2e2 + λ3e3 b = γ 1e1	+ γ 2e2 +		3e3
( ,. , . . λ1 = γ 1,			λ2 = γ 2 , λ3 = γ 3 .
r	|| b . , /			"
! a = (ax , a y , az ); b = (bx ,by ,bz ) a

r

. " α , " a = αb , * * /

,

(ax , a y , az )= (αbx ,αby ,αbz ),

a x = αbx , a y = αby , a z = αbz

	a	x	=	a y	=	a	z	.	(6)
	bx			by		bz
5 ( " (
.

3$". (6) " * , "

( ,, ,

. ' * ", "

( ,.

- $ . !, :

)

r

= (2;−1;3);

b = (3;− 3 2;9 2);

a

)

r

= (2;0;1);

b = (2 3;0;1 3);

a

)

r

= (− 1;3;5);

b = (0;3;0);

a

)

r

= (0;1;0);

b = (0;3; 0).

a

2

− 1

3

r

r

/: )

=

=

a

|| b ;

3

− 3

9

2

2

) ,

2

= 3 =

1

;

2

1

3

3

)

− 1

=

3

,

1

≠

5

. ' ;

− 2 6

2

1

r

) , . . b = 3 a .

!: ), ), ) − ;

) .

2.

* "

r

r

a

, b , c

! ! , c

* *

a

b * "-

* -,.

!

" # !

# ( . 2.13).

13

1 *, *

* * ( " ( . 3 *

$! &$,

" i , j −

*; i , j , k - *.

3. ( #

r r r

(&! # $! &$, #

i , j , k

1)

r

=

r

=

r

= 1,

i

j

k

r

r

r

r

2) i

j ,

i k , j k .

'

(

" ,

*

r r

a

i , j , k . 1

* , - *

. 1 * " , "

	( . (	(	, *
.		!	.
" * , *			.
)	* * ,

" ( * ( * " 0 * " ( " . 1 . 2.14

* *									*, *
	r	r	. *
	e1	, e2
,		*								r r	*
										i , j
r r r	− *.
i , j , k

4. ! $% + # #$ Oxy

* % " * (

14

* * ( Ox, Oy , * ( . ( " , " , * * ( i , j

.

! $% + # #$ Oxyz *

% " * ( * * (

Ox, Oy,Oz , * ( . ( " , " , * * ( i , j , k .

'( (* )

, !&$ !&$ .

* * *

* ( . 2.15, ), * ( .2.15, ).

! Ox #+) #, ##, Oy – #+) ,

Oz – #+) ; " – " .

0 *

*

, " M

OM ( . 2.16).

(5), OM

*

r

r v

i , j , k :

r

r

OM = xi

+ yj + zk .

(7)

%, *

( . 2.16

:

15

OMM ′ − OM = OM ′ + OC ,

OAM ′ − OM ′ = OA + OB .

'

		OM ′ = OA + OB + OC .				(8)
3 OA || i , . " x				, " OA = xi . & "
(	OB = yj ;	OC = zk . !		/	*
OA, OB, OC (8),		* " (7).	6	,	"	(7) x = !0 x OM ;
y = !0 y OM ; z = !0 z OM .
3		, , "	*

Oxyz " * " " (x; y; z ), (

/ ". 1, " (x; y; z ) *

, " *.

3 " M OM . ! /	" (x; y; z )
OM ,	-

" M (OM = (x; y; z )).

3 " (x; y; z ) $% + &$ $

" .

OMM ′ OAM ′

( * (

," , " (*

!)

=

r

=

OA

x

i

x

OM

=

OA

2

+

OB

2

+

OC

2

=

=

OB

=

y

,

OC

=

z

= x2 + y 2 + z 2 .

, -

	OM	= x2 + y 2 + z 2 .			(9)
% * a , * *
OM , ( . 2.17):

16

r

= a x i + a y

j + az k ,

a

r

= a 2x + a 2y + a z2 .

(10)

a

2 AB * " &,

" '

" A(x1 ; y1 ; z1 ) B (x2 ; y2 ; z2 ) ( . 2.18), (

AB " ,.

" :

AB = (x2 − x1 ; y2 − y1 ; z2 − z1 ).

(11)

AB

AB /

"

AB

=

(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2

.

(12)

"

A(x1 ; y1 ; z1 )

B(x2 ; y2 ; z2 )

*

AB (

BA ). "

ρ (A; B):

ρ (A; B) =

AB

=

(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2

.

(13)

, " , " *

" * " " (x; y; z ) - /

	. 4	* , "
,		* * . *
		17

" ( "

" " , * " ":

r	+ b = (ax	+bx , a y	+ by , az + bz ),
a

αa = (αax , αa y , αaz ).

5 * , ( *

" , * *

* .

3 , * * V3 " (

. . * – * * " (

". 2 " , R 3 , .$&$ 2$" #$

# # !$. 3 " (x; y; z ) R3 , " / *, .$&$ 2$" #$ !$

" , x = (x; y; z ).

& " * R 2

* " ( * " (x; y )

(" *) * R − * ( " (")

. * .

'

* (

R 3 , R 2 , R

"

r

1 2

" , "

x

= (x2 + y 2 + z2 )

*

(13).

- $ . 3 ABC

(

- :

A(2; −1; 3);

B(0; 2;4);

C (− 2; −1; 0).

1

AB, − 2BA, AC .

/. '

(11)

AB = (0 − 2; 2 − (− 1);

4 − 3) = (− 2;3;1);

A = (− 2 − 2; − 1 − (− 1); 0 − 3) = (− 4;0; − 3).

% − 2BA = 2(− BA)= 2 AB = 2(− 2;3;1) = (− 4;6;2).

3 * . !

AB AC , &', &) BC ( *

(10):

AB = (− 2)2 + 32 + 12 = 14 ;

AC = (− 4)2 + 02 + (− 3)2 = 5 .

B = (− 2 − 0)2 + (− 1 − 2)2 + (0 − 4) = 29 .

!: AB = (− 2; 3;1;); − 2BA = (− 4;6;2);

AC = (− 4; 0;−3;); AB = 14; AC = 5; BC = 29 .

18

1.6. / !&. ( "

r

3 ". ' * ABCD ( . 2.19) a = AB b = BC . 1 BD, EA, FC , F − AB , E −

DC .

/.

'

BD

* . !

*

*

BA BC :

r

BD = BC + BA = BC + (−

AB )

= b − a .

) ,

: BD *

AB

AD(AD = BC )

* * ,

r

BD = AD − AB = b − a .

r

− b ,

& " EA = ED + EF = −1 2 AB + (− BC )= −1 2a

FC = FB + BC = 1 2 AB + BC =

r

+ b .

1 2a

r

1 r

r

1 r

r

EA = −

− b ;

FC =

+ b .

!: BD = b − a

;

a

a

2

2

r

3 ".

'

= (3;2;−1)

*

"

A(− 1;2;4). 1

a

r

= AB , a0 a .

" B − a

/. ! " B x; y; z . '

(9) * "

(3,2,−1) = (x − (− 1), y − 2, z − 4),

,

*

* ,

(

,

3 = x + 1,

2 = y − 2 , − 1 = z − 4 x = 2,

y = 4,

z = 3.

! " " - . 1

a 0 .

!

r

a

a y

r

a

a

a

=

32 + 22 + 12

a0 =

r

=

rx

,

r

,

rz

,

a

a

a

a

a0 =

3

2

,−

1

,

.

0

14

14

14

!: B (2;4;3),

3

2

1

a0 =

,

,−

.

14

14

14

19

3 ". ',	s	r	= (− 7;6)
	a	= (− 1;2); b = (2;0);c

*. 2 .,

* .
/.	,			*
, . .		r		r	* .
		a	, b , c

4 * , ( . ! ,

, a b . ! * ,

/

r

+ λ2b = 0 *

" λ1a

( λ1

λ2 . !

, /.

λ1 (− 1,2)+ λ2 (2,0) = (0,0)

(− λ1 + 2λ2 , 2λ1 + λ2 0) = (0,0),

− λ1 + 2λ2 = 0,

2λ1

= 0.

5

- :

λ1 = 0 ,

λ2

= 0 . 0 ",

r

,

. . ,

*.

a, b

r

r

r

,

α β −

* c * {a, b }. !

c

= αa + βb

",

* .

* ,.

! *-

,

,

r

r

+

βb

c

= αa

(− 7,6) = α (− 1,2) + β (2,0) (− 7,6) = (− α + 2β ,

2α ),

* "

− α + 2β = −7,

2α

= 6.

- , (

α = 3,

r

r

β = −2. , c = 3a − 2b .

r

r

r

r

!: a, b ;

c

= 3a − 2b .

1.7. 3 " #$# +

&

r

r

'

' ABC AB = a ,

BC = b , AC = c .

r

r

* ,.

AM

, BN , CL .

" a, b , c ,

20