Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Национальный горный университет

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

Высшая математика. Том 2

.pdf

Скачиваний:

Добавлен:

06.06.2015

Размер:

2.75 Mб

Скачать

☆

<<< < Предыдущая 1 23 / 183 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 > Следующая >>>

5. 6 E < ! ! 3 > > '$ 3 - DD ' & D. M$, !, A=B=D=0. " " ! % # < @ " Cz=0 A z=0. /" ! % ! > ' 3 ! 3 ! ' +"#Oy, A " " 3 < & ! % & xOy. @ 3, x=0 –

" " 9 ! % yOz y=0 – ! % xOz.

)

1: + " " ! %, % ! > ' ! ' + Oy,

3 3 M1(1; 0; –1), M2(–1; 2;0).

N + ' +' Oy ! ' ! % #, " " ! % A& Ax+Cy+D=0. > &D3, % M1 α , M2 α , ! + # E > - 3 " " C # + + #& > $> " ' ' # # #

A − C + D = 0,			C = 2D,
	− A + D = 0.	A
			A = D.
	= D=1, $# A=1 C=2. C, " " ! % # < -
@ " x+2z+1=0. ! ': x+2z+1=0. O
			2: + " " ! %, %	! > ' 3 3 &
M(2;3;–4) ! ' ! % yOz (! ! & " +				).
	N + ' ! % yOz\|\|α , " " ! % A& Ax+D=0. ? -
= @ A & M α ,			#& 2A+D=0, D=–2A. #& " " ! % x–2=0. O

1.16. ! % ' 0. ) #% !

@ " # ! % &, % ! < + + $ ! >-' 3 ! 3 . M$ ! % + 9# @ ' # " "# Ax+By+Cz+D=0, C 6 E < $ & D.

# E " " Ax+By+Cz = -D. # C & + '

–D ! = # $@ & @ "	x		+		y		+	z		= 1.
–D ! = # $@ & @ "	− D		+	− D			+	− D		= 1.
		A				B			C
, " % ! 3	a = − D				A	,
					A

b = − D B , c = − D C , ! > # " "

x + y + z = 1. a b c

/ " " < '+" " "# ! % & >.

?'"+&<# @ # 3 3 " 3 + , b . % ! # y=z=0, " " x=a. A #& " D ' "< 3# (0; 0; 0).

+. 1.15. %," "# & >

C, – E C , % + < '+" ! % D + . - @ 3, # C ! , % b – C , % + D '+" D ! % D +"> Oy Oz (+. 1.15). " "# ! % & > &3 - + & +" " ! A& ! %.

)

1: A& & ! % & 2x+3y+6z–6=0.

N ? # E " " " " ! % & >: x + y + z =1 3 2

(+. 1.16). O

2: A& & ! % & 2x–y–4z–4=0.

N @ " # % +! + A ! A& ! %. " ! A& ! % + ' $ A& '-" 99 3, % C ' $ ! "#$.

+. 1.16. %	+. 1.17. % 2x–y–4z–4=0
2x+3y+6z–6=0

$&3 = 3 3 ! & ! % +"# (+. 1.17). O

3: A& & ! % & 2x+5z–10=0.

N % ! ' + Oy. ?$# 3 ! & +"# Oz

(+. 1.18). O

+. 1.18. % 2x+5z–10=0

4: A& & ! % & 3x+2y=0, " ! > ' 3 +' Oz

( . +. 1.19).

5: A& & ! % & 2z+5=0. z=–5/2 ( . +. 1.20).

+. 1.19. % 3x+2y=0

+. 1.20. % 2z+5=0

1.17. 4% 5 !

@ " # ! % α1 α2 , ! " "#:

α1 : A1x + B1 y + C1z + D1 = 0 ,

α2 : A2 x + B2 y + C2 z + D2 = 0 .

& # # C # ! % # # # @ > & , & > E # ! % # (+. 1.21). ? , % & # C # ' # -

! % α1

α2 $ #& > + C > @ >

(

)

(

)

(

)

ϕ =

n1;n2

ϕ = π −

n1;n2

. #&

cos

n1;n2

. + '

A1 A2 + B1B2

+ C1C2

n1 = (A1, B1,C1 )

n2 = (A2 , B2 ,C2 ),

cosϕ = ±

+ B2

+ C2

A2 + B2

+ C2

B D = − Ax0 − By0 − Cz0 >-

&< + A D 3 &, 3 " & ! > ' -

! % , ! A3 + & # C !-

% # A &3 + ' A3 + ">, + '-

! ' @ ! + " ! % -

3 & # C # # '+". +. 1.21. B& # C ! % #

)

3 & # C ! % # x+2y–2z–7=0 x+y–35=0. N ? D % 6 #& D C #:

cosϕ =	A1 A2 + B1B2 + C1C2	=	1 1 + 2 1 − 2 0			=	3	=	2	.
cosϕ =	A12 + B12 + C12 A22 + B22 + C22	=	1 + 4 + 4 1 + 1 + 0			=	3 2	=	2	.
	! ': ϕ = arccos			2	= 45 . O
	! ': ϕ = arccos			2	= 45 . O
				2

1.18.% & ' ( 1.4

& ! ’" & > C3 3, A = 99 + 9# # # & C& @ &!.

?$ & # C ! % #.

1.	x − 3y + 5 = 0, 2x − y + 5z −16 = 0.
2.	x − 3y + z − 1 = 0, x + z −1 = 0.

3. 4x − 5y + 3z − 1 = 0, x − 4 y − z + 9 = 0. 4. 3x − y + 2z + 15 = 0, 5x + 9 y − 3z − 1 = 0.

5. 6x + 2 y − 4z + 17 = 0, 9x + 3y − 6z − 4 = 0.

6. x − y 2 + z − 1 = 0, x + y 2 − z + 3 = 0.

7. 3y − z = 0, 2y + z = 0.

8.6x + 3y − 2z = 0, x + 2 y + 6z − 12 = 0.

9.x + 2 y + 2z − 3 = 0, 16x + 12 y −15z − 1 = 0.


10.	2x − y + 5z + 16 = 0,			x + 2 y + 3z + 8 = 0.
11.	2x + 2 y + z − 1 = 0,		x + z − 1 = 0.
12.	3x + y + z − 4 = 0,		y + z + 5 = 0.
13.	3x − 2 y − 2z − 16 = 0,			x + y − 3z − 7 = 0.
14.	2x + 2 y + z + 9 = 0,		x − y + 3z −1 = 0.
15.	x + 2 y + 2z − 3 = 0,		2x − y + 2z + 5 = 0.
16.	3x + 2 y − 3z − 1 = 0,			x + y + z − 7 = 0.
17.	x − 3y − 2z − 8 = 0,		x + y − z + 3 = 0.
18.	3x − 2 y + 3z + 23 = 0,			y + z + 5 = 0.
19.	x + y + 3z − 7 = 0,		y + z −1 = 0.
20.	x − 2 y + 2z + 17 = 0,			x − 2 y −1 = 0.
21.	x + 2 y − 1 = 0,	x + y + 6 = 0.
22.	2x − z + 5 = 0,	2x + 3y − 7 = 0.
23.	5x + 3y + z − 18 = 0,			2y + z − 9 = 0.
24.	4x + 3z − 2 = 0, x + 2 y + 2z + 5 = 0.
25.	x + 4 y − z + 1 = 0, 2x + y + 4z − 3 = 0.
26.	2 y + z − 9 = 0,	x − y + 2z − 1 = 0.
27.	2x − 6 y + 14z −1 = 0,			5x − 15y + 35z − 3 = 0.

28.	x − y + 7z −1 = 0, 2x − 2 y − 5 = 0.
29.	3x − y − 5 = 0, 2x + y − 3 = 0.
30.	x + y + z 2 − 3 = 0,	x − y + z 2 −1 = 0.
31.	x + 2 y − 2z − 7 = 0,	x + y − 35 = 0.

1.19. 6 & %

0 !. 2 & + !

! % α 1 α 2 ! ' ' , 9> # ' -

! ', 3 '

C, ! % ! #&

$ ' , 6-

E < ! ! > > ! ! E$ :

α1

A1 = kA2 ,

B1 = kB2 , C1 = kC2 .

B2 C2

! % ! ! & "

$ ' , 9> # ' -

! ! & ", C,

= 0 A

A1 A2 + B1B2 + C1C2 = 0 .

# 3 #, α1 α1 A1 A2 + B1B2 + C1C2

= 0 .

M(x0,y0,z0)

α , 9 + 9# @ ' # -

" "#

Ax + By + Cz + D = 0 > # 6 #& D:

d = Ax0 + By0 + Cz0 + D . A2 + B2 + C2

& ' & 6 #& < A+ D 3 " + . ? + +& " 6 #&-A # & " ' + 3 ! + & + – ! D - + D, =$ @ #. + E > 3 C ' ! + - + 9 ! %.

)

1: + " " ! %, % ! > ' 3 3 & M(– 2; 1; 4) ! ' ! % 3x+2y–7z+8=0.

N " " ! % =& # # & @ " Ax+By+Cz+D=0. ?

! ! % > ', %:	A	=	B	=	C	. #& # C ! + A=3, B=2,

	3 2 −7

C=–7. #& " " ! % !$# < @ " 3x+2y–7z+D=0.

B # @, + ' M(–2; 1; 4) α , –6+2–28+D=0, D=32.

C, =& " " 3x+2y–7z+32=0. O

2: + " " ! %, % ! > ' 3 3 M1(1; 1; 1), M2(0; 1; –1) ! ! & " ! % x+y+z=0.

N + ' M1 α , + &D3 " " ! %, % ! >-

' 3 &	3 &, # # # A(x–1)+B(y–1)+C(z–1)=0. , + ' M2
α , ! + =		3 & ! + " ", C # " "
–A–2C=0 A A+2C=0.
>&<#,	%	! % ! ! & " =&$. #&

A+B+C=0.

# 6 E < ! B 3 ": A=–2C, B=C ! + # 9> & ! 3 -

" ": –2C(x–1)+C(y–1)+C(z–1)=0.

+ 3 C&<# –2x+y+z=0. O

3: + " " ! %, % ! > ' 3 3 & M(– 2; 3; 6) ! ! & " ! % # 2x+3y–2z–4=0 3x+5y+z=0.

N + ' M α , A(x+2)+B(x–3)+C(z–6)=0.

	3 α α1 , α α2 ,		#&
	2A + 3B − 2C = 0	2A + 3B = 2C		2A + 3B = 2C	A = 13C

	3A + 5B + C = 0	3A + 5B = −C		B = −8C
	C, " " ! % !$# < @ " 13x–8y+z+44=0. O
	4: ?$ + '		3 M0(1;–1;2) ! %, % ! >-
' 3 3 M1(1;5;–7), M2(–3;6;3), M3(–2;7;3).
	N 3 # α – ! % &, % ! > ' 3				. M1, M2, M3. !&+-
#	3 M0 ! ! & " ! % α C #				3 & M(x; ; z). ? -
" >	3 A <#,	!, #! MM1, M2M1 M3M1.

! + A> > + > #! + ' > ! = # 3, " # " @ + &C ' E > , ! "<# $@

& D. A > # " " ! %,

% ! > ' 3 3

(x1; y1; z1 ), M2 (x2 ; y2 ; z2 )

M3 (x3; y3; z3 ).

x − x1

y − y1

z − z1

x2 − x1

y2 − y1

z2 − z1

= 0 .

x3 − x1

y3 − y1

z3 − z1

+ = ! 3 # <#

x − 1 y − 5 z + 7

−3 − 1 6 − 5 3 + 7

= 0

−4

= 0 .

−2 − 1 7 − 5 3 + 7

−3

A3 + D<# 3 "# ! ! = #& " &:

(x − 1)

1 10

− ( y − 5)

−4 10

+ (z + 7)

−4 1

= 0 .

2 10

−3

−3 2

−10 (x − 1) + 10 ( y − 5) − 5 (z + 7) = 0 .

−2 (x − 1) + 2 ( y − 5) − (z + 7) = 0 . −2 x + 2 + 2 y − 10 − z − 7 = 0 .

? > # " " ! % @ " & Ax + By + Cz + D = 0 .

−2 x + 2 y − z − 15 = 0 .

? > # + ' 3 M0 ! % L:

d = (−2) 1 + 2 (−1) + (−1) 2 − 15 = −21 = 21 = 7 .O
(−2)2 + 22 + (−1)2	9	3

1.20.% & ' ( 1.5

& ! ’" & > C3 3, A = 99 + 9# # # & C& @ &!.

?$ + ' 3 M0 ! %, % ! > ' 3 3

M1, M2, M3.	(1, 5, −4), M3 (−5, −2, 0), M0 (−12, 7, −1).
1. M1 (−3, 4, −7), M2

2. M1 (−1, 2, −3), M2 (4, −1, 0), M3 (2, 1, −2), M0 (1, −6, −5). 3. M1 (−3, −1, 1), M2 (−9, 1, −2), M3 (3, −5, 4), M0 (−7, 0, −1). 4. M1 (1, −1, 1), M2 (−2, 0, 3), M3 (2, 1, −1), M0 (−2, 4, 2).

5. M1 (1, 2, 0), M2 (1, −1, 2), M3 (0, 1, −1), M0 (2, −1, 4). 6. M1 (1, 0, 2), M2 (1, 2, −1), M3 (2, −2, 1), M0 (−5, −9, 1). 7. M1 (1, 2, −3), M2 (1, 0, 1), M3 (−2, −1, 6), M0 (3, −2, −9).

8. M1 (3, 10, −1), M2 (−2,					3, −5),		M3 (−6,	0,	−3), M0 (−6,		7, −10).
9. M1 (−1, 2, 4), M2 (−1, −2, −4), M3 (3, 0, −1), M0 (−2, 3, 5).
10.	M1 (0,	−3,	1), M2 (−4,		1,	2),	M3 (2,	−1,	5), M0 (−3,		4, −5).
11.	M1 (1,	3,	0),	M2 (4,	−1,	2),	M3 (3,	0,	1),	M0 (4,	3,	0).
12.	M1 (−2,	−1,	−1), M2 (0,		3,	2), M3 (3, 1, −4), M0 (−21, 20, −16).
13.	M1 (−3,	−5, 6), M2 (2,			1,	−4),	M3 (0,	−3, −1),		M0 (3,	6,	68).
14.	M1 (2,	−4,	−3), M2 (5,		−6,	0),	M3 (−1,	3, −3),		M0 (2, −10, 8).
15.	M1 (1,	−1,	2),	M2 (2,	1,	2),	M3 (1,	1,	4),	M0 (−3,	2,	7).
16.	M1 (1,	3,	6),	M2 (2,	2,	1),	M3 (−1,	0,	1),	M0 (5,	−4, 5).
17.	M1 (−4,	2,	6), M2 (2,		−3,	0),	M3 (−10, 5, 8),			M0 (−12, 1, 8).
18.	M1 (7,	2,	4),	M2 (7, −1,		−2),	M3 (−5,	−2, −1)		, M0 (10, 1,		8).
19.	M1 (2,	1,	4),	M2 (3, 5, −2),			M3 (−7,	−3, 2),		M0 (−3, 1, 8).
20.	M1 (−1,	−5,	2),	M2 (−6,	0,	−3),	M3 (3,	6,	−3),	M0 (10,	−8,	−7).


21.	M1 (0,	−1,	−1), M2 (−2,			3,	5),		M3 (1, −5,				−9), M0 (−4,		−13, 6).
22.	M1 (5,	2,	0),	M2 (2, 5,		0),		M3 (1,		2,		4),		M0 (−3, −6, −8).
23.	M1 (2,	−1,	−2), M2 (1,			2,	1),		M3 (5,		0,		−6), M0 (14,		−3, 7).
24.	M1 (−2,	0,	−4), M2 (−1,			7,	1),		M3 (4,		−8,		−4), M0 (−6,		5,	5).
25.	M1 (14,	4,	5), M2 (−5,		−3, 2),				M3 (−2,		−6,		−3), M0 (−1,		−8, 7).
26.	M1 (1,	2,	0),	M2 (3, 0,		−3),		M3 (5,		2,		6),		M0 (−13,	−8, 16).
27.	M1 (2,	−1,	2), M2 (1,		2, −1),				M3 (3,		2,		1), M0 (−5,		3,	7).
28.	M1 (1,	1,	2), M2 (−1,		1,		3),		M3 (2,		−2,		4), M0 (2,		3,	8).
29.	M1 (2,	3,	1),	M2 (4, 1,		−2),			M3 (6,		3,		7), M0 (−5,		−4, 8).
30.	M1 (1,	1,	−1), M2 (2,		3,		1),		M3 (3,		2,		1), M0 (−3,		−7, 6).
31.	M1 (1,	5,	−7),	M2 (−3,		6,	3),		M3 (−2,			7,		3), M0 (1,	−1,	2).

1.21. ) . 2

/. ) + /

	=& "					! "# 9 &			! +		E # 3 < '+" =& "#
A& '-" 9 6 + 9
A& '-" 9 6 + 9						3 %1 s , ! ' @ E$ ! "#$.

	s , ! '$ ! "#$, < '+" +! "# &D3 # # E <9
! "# 9.											3 & %1(x1, y1, z1), % =
	C, >$ ! "# L ! > ' 3										3 & %1(x1, y1, z1), % =

! "#$ L ! ' s									= ms	+ ns	+ ps = (m, n, p) (+. 1.22).
	@ " # ' &							3 & %(x, y, z) ! "#$. ? # D 1.22 , %

OM = OM			+ M	M .
		1	1					,			#& $ '+" 3 + t, %
				M	M		s	,			#& $ '+" 3 + t, %
				1
M1M	= ts ,		# C t # C !$# A& '-" 3 + 3 " C

=& " 3 M ! "#$. C t < '+" ! # #. 3-
= &+-
= &+-			3 %1 % ! 3 r		= OM		r	= OM , C&<#
				1		1
r	= r1	+ ts . / " " < '+" # " "# ! "# 9. ! &<,
% C #& 3 D ! # t ! < &+- " 9									3 %, -
= 9 ! "#$.
		? ! = # E " " $ 6 #. # #, %
		? ! = # E " " $ 6 #. # #, %							r = (x, y, z),
			x = x1 + tm,
			(tm,tn,tp), + y = y1 + tn,
r1	= (x1, y1, z1 ) ts =		(tm,tn,tp), + y = y1 + tn,
				1
			z = z + tp.

	+. 1.23. "# 99	+! "#-
+. 1.22. "# & ! +	&D3$ & ! +
C " " D '+" ! # 3 # " "# ! "# 9.
# ! # t # DD '+" x, z,			3 % ! +&-
< '+" C ! "# 9.
1.22. 4 + /
M$ %1(x1,y1,z1) – 3, % = ! "#$ L,
M$ %1(x1,y1,z1) – 3, % = ! "#$ L,		s	= mi	+ n j + pk –

99 +! "# &D3$ . '# # ! "#$ ' & 3 & %(x,y,z) @ " #
	− x1, y − y1, z − z1 ) (+. 1.23).
M1M = (x

	M1M	s ,				#& 9> ! ! A&-
! ! E$ , C,		x − x1	=	y − y1	=	z − z1	– 3 " " ! "# 9.
		m
				n		p

& ' 1. # #, % 3 " " ! "# 9 # C A& - C ! # 3 >, D3 = ! # t. $+, ! # 3 > -

" ' C&<#

x − x1

= t ,

y − y1

= t ,

z − z1

= t A

x − x1

y − y1

z − z1

)

?$ " " ! "# 9

x − 2

y + 1

z − 1

& ! # 3 #& @ ".

−1

N 3 #

x − 2

y + 1

z − 1

= t , + x=2+3t, y=–1+ 2t, z= 1–t. O

−1

& ' 2. M$ ! "# ! ! & " $ > +$,

!, +

! ! & "$ ,

. +! "# &D3$ ! "# 9 s

C, m=0. C, ! # 3 " " ! "# 9 !$#& ' @ ":

x = x1

y = y1 + tn

z = z + tp

x − x1

& ',

D3 D3 " ' ! # t, C # " " ! "# 9 & @ "

	x − x1 = 0
		= z − z1
y − y1

	n		p

# # +" 6 # ' ! +& 3 " " ! "# 9 & @ " = y − y1 = z − z1 . # 3 #, " % # & @ A + 9 '

E 3 <, % ! "# ! ! & " !$ $ +.

@ 3, 3 # " "# &	x − x1	=	y − y1	=	z − z1	! <

0		0			p


! "# ! ! & " +"#	Oy A ! ' + Oz.
	)
1: + 3 $ ! # 3 " " ! "# 9, % !-
> ' 3 3 & %1(1;0;–2) ! ' &
> ' 3 3 & %1(1;0;–2) ! ' &			s = 2i		− 3 j .
N B 3 " " # # @ "		x − 1	=	y	=	z + 2	.
N B 3 " " # # @ "			=	−3	=		.
	2			−3	0
	x = 2t + 1
# 3 " "
# 3 " "	y = −3t .O

	z = −2

2: + " " ! "# 9, % ! > ' 3 3 %1(– 2;1;3), %2(–1;3;0).

N # 3 " " ! "# 9. " E' @ $# +! "# &D-
			= (1,2, −3). L:	x + 2		y − 1		z − 3
3$	s	= M1M2			=		=		.O

			1		2			−3

1.23. 3 . & /,

/ % 0 !

C & ! "#& ! + ! > ' A 3 ! %. F& '-" >, ! D3 +', 3 D ' 99 ! + . C, " " A& '-" > > - > ! %, % @ " D '+" +! ', < " "# E <9 ! "# 9.

@ A& '-" ! ' ! %, @ ' # "-

A1x + B1 y + C1z + D1 = 0,

+ + + =

A2 x B2 y C2 z D2 0

3 D ' ! "#& 9> ! &. / " " D '+" @ ' # " "# ! "# 9.

<<< < Предыдущая 1 23 / 183 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 > Следующая >>>

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]

#
06.06.2015423.42 Кб31Вправи іст. ек.doc
#
06.06.2015774.14 Кб62ВПРАВИ МІКРО.doc
#
06.06.2015293.38 Кб10ВСТУПИТЕЛЬНАЯ ЛЕКЦИЯ №1 по ОТО_2014-2015.doc
#
16.09.2019219.14 Кб5Высшая геодезия.doc
#
06.06.20151.61 Mб24Высшая математика. Том 1.pdf
#
06.06.20152.75 Mб21Высшая математика. Том 2.pdf
#
28.03.2016942.19 Кб12Вышка.pdf
#
06.06.2015270.34 Кб3Гірничий закон України.doc
#
06.06.20151 Mб291Гелогия МПИ.doc
#
06.06.2015186.37 Кб24Генетика.doc
#
15.11.20193.33 Mб21гл1.doc