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Национальный минерально-сырьевой университет «Горный»

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AB AB . -

AB | a |.

. ! -

" . # $ .

, & $ . ' "

0 , ,

" , - & (.

( ) $ -

a b , $ (!). -

r	b , !	r	b .
a		a

, ! " ($ a $+ "

$ ! , $ ! a $ ( a ).

a e " , "

a ( . 2.1).

$ , " " ! "

( . 2.2).	- a b											r
( . 2.2).	- a b											a \|\| b .

									a
					a				a
													a
	e


										b


														b

		3. 2.1							3. 2.2			3. 2.3
		3. 2.1							3. 2.2			3. 2.3
			2							-
-			2					-				c			$
-								-				c			$
-								,			-				-
					r	b
					r
		a			a			"				b			"	!
															( . 2.3).


							B				-					-
				b			B				-					-
-				b					$			a			,
-									$			a			,
			3. 2.4					$+				3. 2.5			-
$			3. 2.4									3. 2.5			-
$															-
								-							.

/ a b . ! " " ! O

OA = a OB = b. .

!" a b ( " ! ,

! OA OB . 6 !

r	b	r	o	( . 2.4).
a	b	, 0 a	b 180	( . 2.4).
				r	b = 90o .
		" # $ a b , a			b = 90o .

$ " a , b , c ,

"( " a b & c , -

+ " ( . 2.5).

" .

% &

1. : !

a b , "

a & b , , & a b -

$ (« » . 2.6). =

, a b !

, + a b , & -

! " ( " (« » . 2.7).

: " ! :

r		r	( ),
1. a	+ b = b + a		( ),
r		r	r	r
2. (a + b) + c = a				+ (b + c) (&),
r	r	(+ ),
3. a	+ 0 = a	(+ ),
r	r
4. a	+ ( a) = 0 .

' #( a b d " a

b , ! b ( . 2.8):	r	r	+ ( b)
	d = a	b = a

			a				- b
			a
		b			r
a			r		a	b
					a	b
				+ b
			a	+ b
							a
			b

		3. 2.6					3. 2.8
	a + b	3. 2.6			3. 2.7		3. 2.8
	r

3. 6 !

$' " a 0 ,

| | | a | , " a , > 0, !-

", < 0. = a = 0 = 0, , $, -

. / a -

a .

: " ! :

1. ( + )a = a + a ( ! ),

r	r	+ b	( ! ),
2. (a	+ b) = a

3. ( a) = ( )a (&),

4.1 a = a (! &).

: " " & " $ -

! ", ! + B &, ! ,

$ B " .

#( , " -		a
#( , " -
! ",
C , ,	A		B	l
C , ,	A		B	l
( . -
-		3. 2.9

+$ e .

/ l a . ! a " A , ! + " , AB= a ( . 2.9). BB B

l .

$& a # l

r		uuur	l
		AB ', AB			,
la	=	uuur
				l
	AB ', AB

AB $+ . - A B', l a =0. -

uuur	r	r
AB = la		e .

: " & ":

1./ & a l $ a -

! a $ l , . .

/la =| a | cos ,

(2.1)

- ! $ l a .

2./ & & ", . .

r	r	+ /l b .
/l ( a	+ b ) = /l a	+ /l b .

3. / ! & ! ! B

, . .

/l ( a) = /la .

$& a b & a$ $ , $ b . / & a b -

r	r	r	b ) .
/br a , ! !, /br a	=\| a	\| cos( a	b ) .

% ) ' # ) ' #

. * '

% & e1 ,e2 ,.....,en

" B B 1, 2,..., n :

1e1 + 2e2 + ... + nen.

H 1, 2,..., n $ BCC & " " &.

% '		e1 ,e2 ,....., en , -
" $ " " &
r	r	r	= 0
1e1	+ 2e2	+ ... + nen

+ BCC & 1, 2,..., n -

" &.

% ' , "

! , 1 = 2 = ... = n = 0 .

+ ' -

. : e1 ,e2 ,.....,en " , , -

! -

" " &

.		a
I B , ! !		a
I B , ! !
$+:	e2
1. "	e2
1. "
, .
, .	O
2. # "	O	e1 3. 2.10
, .		e1 3. 2.10
3. H "
.

* ' -

e1,e2 , ! " .

* ' " e1, e2 , e3 -

, ! " .

, $+

$ $ .

" e1 ,e2 ( . 2.10)

' a ' e1 ,e2 -a

a = xe1 + ye2 .

a $ BCC & x, y B !. # C, a e1 ,e2 x, y,

C a = ( x, y ).

		) " # )
		z			J C -
			M		$ O K
		z			. H B
					. H B
					-
	k		y		i , j k , $+ $
x		O j		y	" . L $ -
x	i	O j		y	&,
	i				$
					$
x		3. 2.11			Ox , Oy Oz . : O
x		3. 2.11			" $ -

" " " ( . 2.11).

) " # a

" Oxyz $ " (x, y, z) ,

x = ir a , y = rj a , z = kr a . N,

r + +

a = xi yj zk .

# C, a ! " " "( x,y, z ), -

: a = ( x, y,z ).

) " # M

" Oxyz $ K - OM ,

OM = xi + yj + zk.

/ B M( x, y,z ) . 2 -

OM , / C, C

uuuur 2 + 2 + 2

OM = x y z .

)(, OM $

, , B Ox , Oy Oz . N-

, x =

cos ,

y =

cos ,

z =

cos ,

cos =

, cos =

cos =

x2 + y2 + z2

x2 + y2 + z2 .

! , (-

, $+ , :

cos2 + cos2 + cos2 = 1.

% & -

/ $ a = (x1; y1; z1) b = (x2 ; y2 ; z2 ) .				: -
$+ !:
1. - $+
, . .
r	; z1	+ z2	) .	(2.2)
a + b = ( x1 + x2 ; y1 + y2	; z1	+ z2	) .	(2.2)
2. - k $ -
$+ B , . .
ka = ( kx1 ;ky1 ;kz1 ) .				(2.3)

I & " C, !

$+ ! (:

1. -" : C " -

a = (x1; y1; z1) b = (x2 ; y2 ; z2 ) ,

, . .

	x1 = x2 , y1 = y2 ,					z1 = z2 .		(2.4)
2.	! & :
A(xA , yA , zA ) B(xB , yB , zB ) ,
	AB = (xB xA; yB yA; zB zA) .							(2.5)
3.	-" : a
b ,
		x1		y1		z1
			=		=		.	(2.6)
		x2		y2		z2

/ B , y2 = 0 , y1 = 0 .

. ) 2.1

1." .

2./ ! : ” $ , ….”

3./ ! : ” $ , ….”

4.H " ? - " & - ! ?

5.- ? - " -

6./ ! : ”& " …...”

7.:C " ! " & ".

8.H " " ?

9./ ! : ” …...”

10." " " -

11./ ! : ” "

$ …...”

12.H $ " ?

13.- " & " C?

2.2.$

/ " $+ -

• ) ' ) .

•. ' ) .

•1 ' ) .

/ C & [2], 1, . 14-28 -$ – $ ! ". / -

(

. : -" " C (

" V 1 V 11-20.

			) '
) ' a b									,
			r		r	:
a b (a,b)						:
			r		r	r			(2.7)
			a	b =\| a \| \| b \| cos( a b ).					(2.7)
						r
= a = 0 b = 0 , $, a b =0.
)
						r	2		r 2
; a								=\| a \| .
=									a = (x1, y1, z1 )
b = (x2 , y2 , z2 ) , !
			r	b = x1x2 + y1 y2 + z1z2 .					(2.8)
			a	b = x1x2 + y1 y2 + z1z2 .					(2.8)
"
1.	r	r	( ),
1.	a	b = b a	( ),
2.	r	r	r	r	r
2.	a	( b + c ) = a b + a			c ( ! ),

3.r r (& !), a ( b ) = ( a b )

4./ : , -

, , -

r	b = 0	x1 x2 + y1 y2 + z1 z2 = 0 .	(2.9)
a

5. a = ( x, y,z ) ! "

+ y2 + z2 .

(2.10)

| a |=

6. - !

a = (x1, y1, z1 )

b = (x2 , y2 , z2 )

r r

x x

+ y y

+ z z

cos(a b) =

(2.11)

x12 + y12 + z12

x22

+ y22

+ z22

|| b

7. : $

&$ , . .

a b =

ar b =

br a .

(2.12)

. '

/ a b ( .

2.12).

. ' a -

" b " c

" :

) c

3. 2.12

a b ! , . .

| c |=| a | | b | sin( a b ) ,

) c a b ,

) "

a,b, c .

a × b .

b = 0 ,

a = 0

! a b ,

0 .

a × b=

" :

b = ( b ×

a ×

a )( ),

! -

2. a ×

( b + c ) = a

b + a ×

c (

b) (&),

a ×

( b) = (a

4. / : , ,

, $

r	× b = 0	r
a	× b = 0	a \|\| b.

5. = a = (x1; y1; z1 ) , b = (x2 ; y2 ; z2 ) , -

! " $+ :

			r	r	k
			r	r
r	r		i	j
r	r	=	x1	y1	z1	.	(2.13)
a	× b	=	x1	y1	z1	.	(2.13)
			x2	y2	z2
6. Y				r	× b + S -
6. Y				a	× b + S -
, a b , ! " :
				r	b \| .		(2.14)
			S =\| a ×		b \| .		(2.14)

1 '

r r

1 ' " " a,b, c -$ , $ a -


b ×	r	. : (		r	r		r r	. =
b ×	c	. : (		(a,b, c)			abc	. =
		r r	$, $		r	r
a,b, c			$, $		(a,b, c) =0.

" ( .

1. / ! " (

, :

r r r r

(a, c,b) = (a,b, c).

2. / ! ! " (

! B , :

r r r r

(a,b,( c)) = (a,b,c).

3. / & " ! " (

, B ! ! ", ,

, , . .

r	r	×	r	×	r	r	( b ×	r	r r
c	( a		b ) = b ( c		a ) = a			c ) = abc .

4. / : ( $

, ! , . .

		r	r			(2.15)
	( a,b ,c ) = 0.					(2.15)
5. /
a = (x1; y1; z1) , b = (x2 ; y2 ; z2 )	c = (x3; y3; z3 ) ,
	r r r	x1	y1	z1
		x1	y1	z1
		x2	y2	z2	.	(2.16)
	(a,b, c) =	x2	y2	z2	.	(2.16)
		x3	y3	z3

6.: ( (a,b, c) Z , -

+	r	c ,
	a,b

r r

$ , " a,b, c

B C

a	c
a
	b

A D

	r r	:
, , " a,b, c		:
r	r		(2.17)
V =\| (a	× b) c \| .		(2.17)
	1
$ 1. ' " ! -
,			a = (2,1,0)

b = (0, 2,1) .

3. 2.13

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