Высшая математика. Том 1

, - .

! AB , , - A′ , -

A′B′ , , , , -

AB A′ ( . 3.2).

. 3.2. #			. 3.3.
		AB
	′
	A

! ABCD, -

, AB = DC AD = BC , AB ≠ AD , BC ≠ DC ,

( . 3.3).

3.3.

$ a ! b , :

1.

b

=

λ

a

;

2. b a ;

3. b

a , ! > 0 ,

! < 0. (! ! = 0, 1 , b

= 0 ).

$ a ! λ a .

1

",

a

, , a ,

2

, , a .

' :

1. $ -

α

β

a

α (β a) = (αβ )a .

$ ,

,

,

-

α

β

a

. ( , , ,

) a , α β ,

β .

a , α

$ -

2. " a ≠ 0 .

b !, b = λ a .

111

* ! , a

, .

"

"

O -

a

b – .

OA = a . # A

AB = b .

OB ,

' a c

b ,

a + b = OA

+ AB .

. 3.4. +

+, -

, :

O OA = a OC

= b ;

#. ' OC

= AB , OB , -

, ) O,

a + b .

. 3.5. #

		- .
		1.	!,	-
										a
				a , a + 0		= a .
		2. $ , a + b = b + a .
		. .
		3. $ , -
					/
a + (b + c) = (a + b) + c .
a + b + c .
		+ . & )
O ,							. $ -

, – . , ' -

112

, . 0 - .

. 3.6. +

4. $ - ! -
			a	b :

λ (a + b) = λ a + λ b .

. 3.7. *

&,	!
,	; , .
'
	a ,	b	a	+ b , ,

!,

, , ,

λ a + λ b = λ (a + b).

5. $ - a b - a

(α + β )a = α a + β a .

$%

,

a ,

, a

−a

a .

. 3.8.

113

					-
	# a
a ! = –1:			−a = (−1) a .
	$% a						b	c = a − b ,
a −b ,				c = a +			(−b).
					+
	&, a − a = a					(−a) = 0 - a .
	- , b + (a					− b) = a .
	/ , c = a					− b , b + c = a .
	& -
										) O.
. OA = a								OB = b

		. 3.9. #
	1 -		,
					OA		−OB
BO .	/ OA + BO = BO + OA = BA.
		BA , ' a , b « ’ -

» « » ( ),
a − b . $ , BO + BA = OA b + c = a .
	/ ,				) O,
			a	b ,
					,		-
OACB, OC
	,
		a	+ b , BA , -
	, a − b .

3.4. ! . "

"

a

b . )

O OA = a

OB = b .

. 3.10. ( a b

114

& a b AOB .
	≤ ϕ ≤ π .
( (a;b) = ϕ , 0

l e ( ,

).

				l
	. 3.11. ( a
			l	ϕ
	# a					a
e . ', l –		a = AB – . # A1 B1 )

l A B. #, A x , B –

1 1 1

x l. / % ' AB l

2

x2 – x1 AB .

. 3.12. # AB l

# a	l a =					AB .
!,				l	l
		a		l , x2> x1,

x2 – x1> 0; , x2< x1 x2 – x1< 0. ", -

a l, x = x x – x = 0.

2 1 2 1

/ , AB l – A1B1,

. ', . 0 . % -

) , -

.	.
1. # a l							a
						cos(ϕ ) .
	: l a =	a		cos(a;l ) =	a

115

. 3.13. # a l,

2.#

( + ) = +

:l a b l a l b .

. # -

.

. 3.14. #

.- -

.

3. ! a ! ,

: l (λ a) = λ l a .

3.5. # ! " # ! "

.

a1, , ak .

( % ' - -

– . * λ1, ,λk -

a = λ1

a1 + λ2 a2

+ + λk ak , λ1, ,λk

, ) ).

a

.

a1, ,ak ,

",

a , b , c , ) ) )

1

a

+ c

) : d

= 3a −

b + c , f

=

, g

= 2a .

2

2

! ,

, .

116

. 3.15. c				a			b
			,
	a , , a

λ1, ,λk , , λ1 a1 + λ2 a2 + + λk ak = 0 . !, -

, -

.
	%					,
	+ λ2					λ1 = λ2			= = λk = 0 , -
λ1 a1		a2	+ + λk ak	= 0
			.

1. 2- ,

.

/ , ,

.

2. / ,

.

/ , . ( , , .

3.6. #, # # $#. % "

. & % $# ' %

)

. 3 e1,e2 , ,ek .

% , ,

, 1 ( - -

, ). ', -

- .

0 -

. ', . + .

		3. "
			e ,e ,e . / - -
			1 2	3
	a	) )		a = xe1		+ ye2	+ ze3 , x,

y, z – . / .

117

. 3.16.	a	e1,e2 ,e3

! ) , :
		, - ,
! e ,e
	1 2
e1,e2	a = xe1		+ ye2 , -

.

. 3.17. a	e1,e2

/ , -

– , -

: a → ( x, y, z). , ) x, y, z -

							,
xe	+ ye	+ ze	= a .
1	2	3
	!		e ,e ,e		a = xe		+ ye + ze , x, y, z -
			1 2	3		1	2	3
					a	.		(	a
a = ( x, y, z) .

3.7. (

" O e1,e2 ,e3 .

" ( )

, -

(2- ), ) .

/ O ; ,

, -

– , . #, -

118

, .

. 3.18. / M e1,e2 ,e3

M. % -

M. OM , ' -

M - M.

OM –

: OM = xe1 + ye2 + ze3 .

( - M

M M(x,y,z).

# , – , –

.

0 . /

– .

. 3.19. $

# . &

, ) ,

.

! , , , -

,

. ' -
i, j, k , Ox, Oy Oz, -
: a = xi	+ y j + zk .

119

# 1. # -
			-		(–1;3) ( .3.20).
a = −i	+ 3 j . a
# 2. # b = 2i				− j + 3k . b -

- N(2; –1; 3) ( .3.21).

	. 3.20. a = −i + 3 j
		. 3.21.	b = 2i	− j + 3k

"

.

3.8.

" a . " *, +, ,

. ( cos *, cos +, cos , -

.

. 3.22. a *, +, ,

120