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Национальный минерально-сырьевой университет «Горный»

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8. ; %

, 2 %

2x 3y 5 = 0,x + 2 y + 1 = 0.

L ,

7 y 7 = 0,

y = 1, x = 2 y 1 = 1.

5: (1,-1).

, 4. O 2x+3y-6=0

2x-3y+7=0

8. I ,

. " , -

, x=0 y=2. T ' (3.16), , -

% n(2,3) , ,=7

d =

| 2 0

+ 3 2 + 7 |

22 + 32

, 5. - (4,5), .(-3,0), ,(-8,3) . O

, . -,.

8. W

O - -, - '

x =

xA + xC

4 8

= 2 y =

yA + yC

5 + 3

= 4 .

" ' (3.13) - , %

.(-3,0) (-2,4).

x + 3

y 0

2 + 3

4 0

. : 4x y + 12 = 0 . 5: . : 4x y + 12 = 0 .

, 6. - (4,6), .(-4,0), ,(-1,-4). O

, . -,.

8. " , % -

., AC . T - ,,

AC( 1 4, 4 6) = ( 5, 10) . " ' (3.10)

5(x ( 4)) + ( 10)( y 0) = 0 ,

x + 2 y + 4 = 0 .

5: x + 2 y + 4 = 0 .

, 7. 6 , % -(2,5) -

x , 7.

8. O , %

– 2 x. ; 2 (7,0). 1

' (3.13):

x 2	=	y	5
				,
7 2		0	5	,

x + y 7 = 0 .

5: x + y 7 = 0 .

, 8. O y x 3 = 0 2x + 3y + 4 = 0 8. ; % % %

: n1( 1,1) n2(2,3) . ' (3.14)

cos =

( 1) 2 + 1 3

( 1)2 + 12

22 + 32

= arccos		1	.

	26				1
	26
5: = arccos						.

				26
				26

-5

9. O %, % x ,

				5,	Oy - , -
				4, %
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4				8.	T
				% (3.9). L,
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			x	, ( . 3.14):

-4

= 1

4x 5 y = 20

7. 3.14

4x 5 y + 20 = 0

4x 5 y 20 = 0

5: % %

4x 5 y + 20 = 0 4x 5 y 20 = 0 .

9 & !' & # & 3.2

1. ; ?

2.; %?

3.; 2'' H k

y = kx + b ?

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5." , .

6." .

7. 0 , % .

8.L % .

9.6' % % .

10." ' % .

3.3. ( # ' &! & ' % # & #

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•! " # $# % &! # & #.

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•(*! ' '.

% "

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[2], 3, . 43-58 – %

6 - ' -

( 21-30.

! " # $# % &! # & #

1. 17 $# &! 17 $# &!

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(3.17)

7. 3.15

A, B C . W ,

(3.17) -

Oxyz

(3.17). 2

n = ( A,B,C ) 2 2 !3-

# &! ( .3.15), 2'' H A , B , C -

2 .

G % %

' , H.

			2. ( # &! # -
	z		K (3.17) 2'' H-
			, ,
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x	a		a = D/A, b = D/B,						c = D/C .

L (3.18) $# 7. 3.16 &! "# -". W , -

a,b, c ,

% ( . 3.16).

3. ( # &!, &-'7 % "		$4 "$ -
!3 $ #$
M0 (x0 , y0 , z0 ) , P ,
2 n = (A, B,C) ( . 3.17).
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M0M = (x x0 , y y0 , z z0 ) ,		P , -

. L M0M n

z n

. 7 3.17

- 2

M0 M

. 7 3.18

		M0M n = 0.
		K -
		, , 2
		', -
	M	P :
	M	A(x x0 ) + B( y y0 ) + C(z z0 ) = 0. (3.19)
		A(x x0 ) + B( y y0 ) + C(z z0 ) = 0. (3.19)
		K -
		r0 r M0 M , %

(r r0 ) n = 0 .

(3.20)

$# &! # # % /-

4. ( # &!, &-'7 % "

$4 "$ & !! !3 #$ !- ! #

O P , %-

M0 (x0 , y0 , z0 ) -

a(a1, a2,a3)	b(b1,b2,b3)	( .3.18). L
M	P	r
M	P	M0M ,a,b , -

% -

( M0M		r
( M0M		, a,b) = 0.
" % % 2
	x x0	y y0	z z0
	x x0	y y0	z z0
	a1	a2	a3	= 0.	(3.21)
	b1	b2	b3

5. ( # &!, &-'7 % " ", ! : 7

% & ' %

1 (x1 , y1 , z1 ) , 2 (x2 , y2 , z2 ) M0 (x0 , y0 , z0 ) , -

( . 3.19). O -

M0 M

. 7 3.19

. I ,

a b , -

, M0 1 M0 2 . "

x x0	y y0	z z0
x x0	y y0	z z0
x1 x0	y1 y0	z1 z0	= 0.	(3.22)
x2 x0	y2 y0	z2 z0

! " # $# % & ' % # & #

1.17 $# ' & ' % # &-

K %

P1 P2 , %

A1x + B1 y + C1z + D1 = 0 A2 x + B2 y + C2 z + D2 = 0 ,

( .3.20), 2

, % 17-

$# ' :

7. 3.20

		l		A x + B y + C z + D = 0,				(3.23)
		l		1	1	1	1	(3.23)
s			A2x + B2 y + C2z + D2 = 0.
s
	M	2. 9-& " $#
		2. 9-& " $#
M0		& ' % # & #
M0		l	- . K
		l	- . K

.3.217

M0 (x0 , y0 , z0 ) , , s = (m, n, p) , 2

( .3.21). " s & #!'47 # & ' %.

K M (x, y, z) - l , M0M s -

. %

t ,

M0M = ts.

K - r0 r M0	M , 2 '-

r r0 = ts.	(3.24)

L (3.24) #-& " -

3. , " " $# ' & ' % # & #

(3.24) ',

% x, y, z & " $# '

x = x0 + mt,
	+ nt,	(3.25)
y = y0
	+ pt.
z = z0

1 %

K %

', " $# ' :

x x0	=	y y0	=	z z0	.
m		n
				p

K - (3.26) ,

4.( # & ' % " # " # &-

, %

M0(x0, y0, z0) M1(x1, y1, z1) ( . 3.22), -

% :

M0M = (x x0, y y0,z z0) M0M = (x1 x0, y1 y0,z1 z0) ,

M (x, y, z) - :

(3.26)

l 0

7 3.22

x x0	=	y y0	=	z z0	.	(3.27)
		y1 y0
x1 x0				z1 z0

		(*! ' '
1. (* ! :$ & ' # & #
(*! :$ & ' l1 l2			, %
l1 l2 .
		l1		l1
	r		r	r
	s1		s2	s1

	r
	s2	l2		l2
		l2		l2
)		.	7 3.23	)
		.	7 3.23

5, %

, ( .3.23). 1,

cos =

(3.28)

| s1 || s2 |

2. (* ! :$ & ' % &!-

34 # & #

(*! :$ & ' % &!-

34 -

H -

( .3.24).

sin = sin(90o ) =

cos

7. 3.24

- -

s . 6,

sin =

n s

(3.29)

| n || s |

3. (* ! :$ &! '

(*! :$ #$' &! ' 21 22

% , -

% 2 .

, -

7. 3.25	.

5, % -

, cos =| cos | , -

( . 3.25). 1

		r	r
		n	n
cos =		r1	r2	.	(3.30)
cos =		r1	r2	.	(3.30)
	\| n1 \|\| n2 \|

4. ' :$ #$' " # & #

7 M0 (x0 , y0 , z0 ) M1(x1, y1, z1 ) -

M1M0 .

;

M1M0 = (x0 x1, y0 y1, z0 z1) ,

uuuuuuur

l =

= (x0 x1)2 + ( y0 y1)2 + (z0 z1)2

M1M 0

(3.31)

5. ' " # & # &!

M0 (x0 , y0 , z0 )

P ,

Ax + By + Cz + D = 0 . S

0 ,

( . 3.26), -

7 3.26

S =

| Ax0 + By0 + Cz0 + D |

(3.32)

+ B2

+ C2

6.' " & ' % # & #

d M0(x0, y0, z0) l -

, ,
s . " M1(x1, y1, z1) ,
	r	M1M0
	r
d =	s ×		.	(3.33)
		r
		r
		\| s \|

8 & #

, 1. O , % A( 1,2,1)

n(2,2, 4) .

8. L (3.19)

2(x +1) + 2( y 2) 4(z 1) = 0.

7 ,

2x + 2y 4z + 2 = 0 x + y 2z +1 = 0 .

5: x + y 2z +1 = 0 .

, 2. O , %

A(1, 1,3) , B(2,0, 1) , C(5,2,1).

8. (3.22):

x 1	y +1	z 3
x 1	y +1	z 3
2 1	0 ( 1)	1 3	= 0.
5 1	2 ( 1)	1 3

% -

:01 x 41 y z 12 = 0.

5: 01 x 41

y z 12

= 0.

, 3. 6 , %

A(2,1,3) B(4,5,3).

8. " M0

A ,

- AB = (2, 4,0) . ;

2 :

x 2

y 1

z 3

x 2

y 1

, z = 3.

x 2

y 1

, z = 3.

, 4.

2x y + 3z + 3 = 0

M0 (1,2,1)

s = (2,4, 2) .

8. :

1+ 2t

2 + 4t.

1 2t

; M (x, y, z)

. x, y z , :

(12 + 2t) (2 + 4t) +(13 2t) + 3 = 6t + 6 = 0.

5 t = 1 , 2

: M (3,6, 1) .

5: M (3,6, 1) . , 5. :

	x y + z 2	=	0
		5 =	.
2x + y 2z +		5 =	0

O 2 .

8. T x y, -

3y 4z + 9 = 0 3x z + 3 = 0 .

" z % -

		x +1		=		y + 3		=		z	.
		1		=				=			.
		1		4				3					x +1		y + 3		z
													x +1	=	y + 3		z
5:																=		.
5:													1		4	=	3	.
, 6. O l1 , -
:
	x 1		=		y 2		=		z 1			,
	5		=		4		=		3			,
	5				4				3

l2 , :

2x y + z + 3 = 0,

x y + z + 5 = 0.

8. O s2 l2 -

% :

1 1

= 0i

% ' (3.28):

cos =

5 0 + 4 ( 1) + 3 ( 1)

52 + 42 + 32

+ ( 1)2

5: arccos (

) .

, 7. O M0(3, 2,1) l1 ,

x 2

y + 5

z 3

8. " ,

, M1(2, 5,3) , M1M0

l s = (2, 3,4)

' (3.33):

			r	r
			r	r
r		uuuuuuur	i	j
r		uuuuuuur
s	×	M1M0 =	2	3
			1	3

= ( 1, 3, 2) . O

. " ,

k	r	r	r
4	r	r	r
4	= 6i	8 j	9k.
2

2 ' (3.33):

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