Высшая математика. Том 1

& .

" , a = xi

+ y j + zk .

x

/ x = Ox a =

a

cosα , cosα =

.

y

a

y = Oy a

=

a

cos β , cos β =

.

a

z

z = Oz a =

a

cosγ , cosγ =

.

a

" , cos2 α + cos2 β + cos2 γ = 1.

" , -

.

3.9.

)

# !

= (λ x,λ y,λ z) .

, a

= xi

+ y j + zk , λ a

= λ xi

+ λ y j + λ zk

#

) ,

+ y

a

= x i

+ y j

+ z k

b

= x i

j + z

k , :

1

1

1

2

2

2

a

+ b = ( x1 + x2 )i + ( y1 + y2 ) j +

( z1 + z2 )k .

* ,.

$ , ) -

.

/,

a

= x1i

+ y1 j

+ z1 k

b

= x2 i

+ y2 j + z2 k ,

x1

y1

z1

a b

=

=

.

x2

y2

z2

# 1. $

= (3;−2;5).

&

a = (2;3;5) b

c = 2a − b .

4 &

2a = 2 (2;3;5) = (4;6;10) .

c = 2a

− b = (4;6;10) − (3;−2;5) = (1;8;5).5

# 2. "
						x
						= {1,1, − 4} .
x = {4,3, − 12} ,	p = {2,1, − 1} , q = {1,1, − 1} , r
4 #			x x = ( x1;
: x = x1		p + x2	q + x3	r

# :

p ,	q		r :

x2 ; x3 ) . /

121

(4;3;−12)

= x1 (2;1;−1) + x2 (1;1;−1) + x3 (1;1;−4) .

&

:

(4;3;−12)

= (2x1;1x1;−1x1 ) +

(1x2 ;1x2 ;−1x2 )

+ (1x3;1x3;−4x3 ).

&

:

(4;3;−12)

= (2x1 + x2 + x3;

x1 + x2 + x3;

− x1 − x2 − 4x3 ) .

. x1; x2 ; x3 -

:

2x1 + x2 + x3 = 4,

x1 + x2 + x3

= 3,

= −12.

− x1 − x2 − 4x3

’ )

x ; x ; x

, , , x -

1 2 3

p ,

q

r :

’ +-0, , (:

∆ =

2

1

1

= −3; ∆1 =

4

1

1

= −3

1 1

1

3

1

1

−1 −1 −4

−12 −1 −4

∆2 =

2

4

1

= 3; ∆3 =

2

1

4

= −9

1

3

1

1 1

3

−1 −12 −4

−1 −1 −12

x =

∆1

=

−3 = 1; x

=

∆2

=

3

= −1; x

= ∆3 = −9 = 3 .

1

∆

−3

2

∆

−3

3

∆ −3

',

x = p

− q + 3r . 5

3.10. + & !

A( x ; y ; z ) B( x ; y ; z ). & -

1 1 1 2 2 2

AB . &, AB = OB − OA.

. 3.23. AB

122

&

( x ; y

; z

) .

OA = ( x ; y ; z ) OB =

1 1

1

− y

2 2

2

AB = x i + y

j + z

k − x i − y j − z k = ( x

− x

)i + ( y

) j + ( z

− z

)k .

2

2

2

1

1

1

2

1

2

1

2

1

/ , AB ,

.

# 1. & A(1; –2; 3), B(2; 0; –1). &

BA .

4 &

BA , -

= (1;−2;3) − (2;0;−1) = (−1;−2;4) .5

: BA

# 2. $ A(–2; 3; 1), B(–1; 2; 0), C(0; 1; 1). & a =

2AB

+ AC .

−1;−1),

= (2;

−2;0) .

4 AB = (1;

AC

(2;−2;0) = (4;−4;

−2) .5

a =

2AB +

AC = (2;−2;−2) +

# 3. , CD

= CA + CB . & D,

#(3; –4; –1), B(–4; 4; 1), C(–3; –5; 4).

4 &

= (5;10;−8) .

CA = (6;1;−5), CB = (−1;9;3), CD = CA + CB

#

D( x; y; z),

CD = ( x + 3; y + 5; z − 4).

', (x+3; y+5; z–4)=(5;10;–8).

& x=2, y=5, z=–4, D D(2; 5; –4). 5

3.11. , " # - 3.1

+ ’ , ))

) .

x

p , q , r .

1.

x = {−2, 4, 7} , p = {0, 1, 2} , q = {1, 0, 1} , r = {−1, 2, 4}.

2.

x = {6, 12,

−1} ,

p = {1,

3,

0} ,

q = {2,

−1,

1} ,

r = {0,

−1,

2}.

3.

x = {1, −4, 4} , p = {2, 1, −1} , q = {0, 3,

2} , r = {1, −1, 1}.

4.

x = {−9, 5, 5} , p = {4, 1, 1} , q = {2, 0, −3} , r = {−1, 2, 1}.

5.

x = {−5, −5, 5} , p = {−2, 0, 1} , q = {1, 3, −1} , r = {0, 4, 1}.

6.

x = {13,

2,

7} ,

p = {5,

1,

0} ,

q = {2,

−1,

3} ,

r = {1,

0,

−1}.

7.

x = {−19,

−1, 7} , p = {0, 1,

1} ,

q = {−2, 0,

1} ,

r = {3,

1,

0}.

8.

x = {3, −3, 4} , p = {1, 0, 2} , q = {0, 1, 1} , r = {2, −1, 4}.

123

9.

x = {3, 3, −1} , p = {3, 1, 0} , q = {−1, 2, 1} , r = {−1, 0, 2}.

10.

x = {−1,

7,

−4} ,

p = {−1,

2,

1} , q = {2,

0,

3} ,

r = {1,

1,

−1}.

11.

x = {6,

5, −14} ,

p = {1,

1,

4} ,

q = {0,

−3, 2} ,

r = {2,

1,

−1}.

12.

x = {6,

−1,

7} ,

p = {1,

−2, 0} ,

q = {−1,

1,

3} ,

r = {1,

0,

4}.

13.

x = {5,

15,

0} ,

p = {1,

0,

5} ,

q = {−1,

3,

2} ,

r = {0,

−1,

1}.

14.

x = {2,

−1,

11} ,

p = {1,

1,

0} ,

q = {0,

1,

−2}

,

r = {1,

0,

3}.

15.

x = {11,

5,

−3} ,

p = {1,

0,

2} ,

q = {−1,

0,

1} ,

r = {2,

5,

−3}.

16.

x = {8,

0,

5} ,

p = {2,

0,

1} ,

q = {1,

1,

0} ,

r = {4,

1,

2} .

17.

x = {3,

1,

8} ,

p = {0, 1,

3} ,

q = {1,

2,

−1} ,

r = {2,

0,

−1}.

18.

x = {8,

1,

12} ,

p = {1,

2, −1} ,

q = {3,

0,

2} ,

r = {−1,

1,

1}.

19.

x = {−9,

−8,

−3} ,

p = {1,

4,

1} ,

q = {−3,

2,

0} ,

r = {1, −1,

2}.

20.

x = {−5,

9,

−13} ,

p = {0,

1,

−2} , q = {3,

−1,

1} ,

r = {4,

1,

0}.

21.

x = {−15,

5,

6} ,

p = {0,

5,

1} ,

q = {3,

2, −1} ,

r = {−1,

1,

0}.

22.

x = {8,

9,

4} ,

p = {1,

0,

1} ,

q = {0,

−2, 1} ,

r = {1,

3,

0} .

23.

x = {23,

−14, −30} , p = {2,

1,

0} , q = {1,

−1, 0} ,

r = {−3,

2,

5}.

24.

x = {3,

1,

3} ,

p = {2,

1,

0} ,

q = {1,

0,

1} ,

r = {4,

2,

1} .

25.

x = {−1,

7,

0} ,

p = {0,

3,

1} ,

q = {1,

−1,

2} ,

r = {2,

−1,

0}.

26.

x = {11,

−1,

4} ,

p = {1,

−1,

2} , q = {3,

2,

0} ,

r = {−1,

1,

1}.

27.

x = {−13,

2,

18} ,

p = {1,

1,

4} ,

q = {−3,

0,

2} ,

r = {1,

2,

−1}.

28.

x = {0,

−8,

9} ,

p = {0,

−2, 1} ,

q = {3,

1, −1} ,

r = {4,

0,

1}.

29.

x = {8,

−7,

−13}

,

p = {0,

1,

5} ,

q = {3,

−1, 2} ,

r = {−1,

0,

1}.

30.

x = {2,

7,

5} ,

p = {1,

0,

1} ,

q = {1,

−2, 0} ,

r = {0,

3,

1} .

31.

x ={15,

−20, −1}

,

p ={0,

2,

1} ,

q = {0,

1,

−1} ,

r = {5, −3,

2}.

3.12. , " # - 3.2

+ ’ , ))

) .

&' c1 c2 , -

a b .

1. a = {1, −2, 3} , b = {3, 0, −1} , c1 = 2a + 4b, c2 = 3b − a.

124

2. a = {1, 0, 1} , b = {−2, 3, 5} , c1 = a + 2b, c2 = 3a − b.

3. a = {−2, 4, 1} , b = {1, −2, 7} , c1 = 5a + 3b, c2 = 2a − b. 4. a = {1, 2, −3} , b = {2, −1, −1} , c1 = 4a + 3b, c2 = 8a − b. 5. a = {3, 5, 4} , b = {5, 9, 7} , c1 = −2a + b, c2 = 3a − 2b. 6. a = {1, 4, −2} , b = {1, 1, −1} , c1 = a + b, c2 = 4a + 2b.

7. a = {1, −2, 5} , b = {3, −1, 0} , c1 = 4a − 2b, c2 = b − 2a. 8. a = {3, 4, −1} , b = {2, −1, 1} , c1 = 6a − 3b, c2 = b − 2a.

9. a = {−2, −3, −2} , b = {1, 0, 5} , c1 = 3a + 9b, c2 = −a − 3b.

10.

a = {−1,

4,

2} ,

b = {3,

−2,

6} ,

c1 = 2a − b,

c2

= 3b − 6a.

11.

a = {5,

0,

−1} ,

b = {7,

2,

3} ,

c1 = 2a − b,

c2 = 3b − 6a.

12.

a = {0,

3,

−2} ,

b = {1,

−2,

1} ,

c1 = 5a − 2b,

c2

= 3a + 5b.

13.

a = {−2,

7,

−1} ,

b = {−3,

5,

2} ,

c1 = 2a + 3b,

c2 = 3a + 2b.

14.

a = {3,

7,

0} ,

b = {1,

−3,

4} ,

c1 = 4a − 2b,

c2

= b − 2a.

15.

a = {−1,

2,

−1}

,

b = {2,

−7,

1}

,

c1 = 6a − 2b,

c2 = b − 3a.

16.

a = {7,

9,

−2} ,

b = {5,

4,

3}

,

c1 = 4a − b,

c2 = 4b − a.

17.

a = {5,

0,

−2} ,

b = {6,

4,

3}

,

c1 = 5a − 3b,

c2

= 6b −10a.

18.

a = {8,

3,

−1} ,

b = {4,

1,

3} ,

c1 = 2a − b,

c2 = 2b − 4a.

19.

a = {3,

−1,

6} ,

b = {5,

7,

10} ,

c1 = 4a − 2b,

c2

= b − 2a.

20.

a = {1,

−2,

4} ,

b = {7,

3,

5} ,

c1 = 6a − 3b,

c2

= b − 2a.

21.

a = {3,

7,

0} ,

b = {4,

6, −1} ,

c1 = 3a + 2b,

c2

= 5a − 7b.

22.

a = {2,

−1,

4} ,

b = {3,

−7,

−6} ,

c1 = 2a − 3b,

c2 = 3a − 2b.

23.

a = {5,

−1,

−2}

,

b = {6,

0,

7} ,

c1 = 3a − 2b,

c2

= 4b − 6a.

24.

a = {−9,

5,

3} ,

b = {7,

1,

−2} ,

c1 = 2a − b,

c2

= 3a + 5b.

25.

a = {4,

2,

9} ,

b = {0,

−1,

3} ,

c1 = 4b − 3a,

c2

= 4a − 3b.

26.

a = {2,

−1,

6} ,

b = {−1,

3,

8} ,

c1 = 5a − 2b,

c2

= 2a − 5b.

27.

a = {5,

0,

8} ,

b = {−3,

1,

7} ,

c1 = 3a − 4b,

c2

= 12b − 9a.

28.

a = {−1,

3,

4} ,

b = {2,

−1,

0} ,

c1 = 6a − 2b,

c2

= b − 3a.

29.

a = {4,

2,

−7} ,

b = {5,

0,

−3} ,

c1 = a − 3b,

c2

= 6b − 2a.

30.

a = {2,

0,

−5} ,

b = {1,

−3,

4} ,

c1 = 2a − 5b,

c2

= 5a − 2b.

31.

a = {−1,

2,

8} ,

b = {3,

7,

−1} ,

c1 = 4a − 3b,

c2

= 9b − 12a.

125

3.13. . $ %

. 0

. 0

, . / -

: .
" a b , ϕ = (ab).
				,
	a		b

. + -

cosϕ .

a b . ',

a b =

a

b

. 3.24. +

! , -

.

.

1. + -

, -

a

b a b = b a .

:

2. $ - ! -

a

b

λ (a

b) =

(λ a) b = a (λb).

3. $ - a,b,c :

(a + b)

c = a

c

+ b

c .

2

2

4. $ - a

a

= a a =

a

.

6 )

=

2

=

a

a

a a = a .

5. +

,

.

. .

/ , $& / " / / % "

) .

0

$ c = 2a + 3b . ,

a

=

4 ,

b

=

.

&

c

.

3, a;b = 60

2

=

2

,

=

2

4 7 c

c

c

c .

126

&:

2

2

2

2

=

& -

c

= (2a + 3b)

4a

+ 12ab

+ 9b .

cosϕ , :

a

b ,

a b =

a

b

2

2

2

c

4

a

b

= 64 + 72 + 81

= 217 .

=

+ 12abcos(a;b)+ 9

',

= 217 . 5

c

3.14. ( ! . ! .

* % " &

, , -

,.

"

a = x i

+ y j + z k

1

1

1

b = x2 i + y2 j + z2 k .

i, j,k . '-

, ) -

, . -

3.1:

/ 3.1

+

i, j, k

# :
		a b =		(x1i + y1	j + z1 k ) (x2 i + y2			j + z2 k ) =
2					2					2
= x1x2 i	+ x1 y2 i j + x1z2 ik		+ y1x2 ji + y1 y2 j			+ y1z2 jk + z1x2 ki			+ z1 y2 k j + z1z2 k		=

= x1 x2 + y1 y2 + z1 z2 .

',

y2 + z1 z2 .

: a b = x1 x2 + y1

. -

:

=

2

= x2 + y2 + z2 .

a

a

1

1

1

$

cosϕ

a b =

a

b

127

cosϕ = a b . a b

) -, , :

	cosϕ =		x1x2 + y1 y2 + z1z2
		x2	+ y2 + z2		x2 + y2 + z2
	1		1	1	2	2	2

* % " ,:
	axbx + ayby + azbz = 0 .
a b a b = 0

/ , -

, -

.

a b ;

# 1. " #(–1; 1; 0), B(3; 1; –2), a

= AB ,

; cosϕ .

a

b

−3) .

4 a = (3 + 1;1 − 1;−2) = (4;0;−2) ;

b = (2;0;

= 4 2 + 0

0

+ (−2) (−3) = 8 + 6 = 14 .

a b

42 + 22

20 = 2 5 ;

22 + 32 =

13 .

cosϕ

a

=

=

b

=

b = 2i	− 3k . &:

=		a		b	=	7	.5
						65
		a		b

# 2. & B ABC ,

A(1; 5; 6), B(5; 3; 10), C(2; 1; 14).

( )

4 =

B BA, BC

				. 3.25. ( BA					BC
		= (1 − 5;5 −						= (2 − 5;1 − 3;14 − 10) = (−3;−2;4).
BA				3;6 − 10) = (−4;2;−4) ; BC
			= (−4) (−3) + 2 (−2) + (−4) 4 = 12 − 4 − 16 = −8 .
BA	BC
			42 + 22 + 42 =		36 = 6 ;		= 32	+ 22 + 42 = 29 .
BA		=				BC

128

cosϕ =

BA

BC

=

−8

= −

4

4 !

0

.

ϕ = arccos

−

#

= 104

.5

6

29

3

29

BA

BC

3

29 %

# 3.

#

m

a = mi +

3 j − k

?

b = 2i

− 5 j + mk

= 2

m − 15 − m = 0

4 & a b

', m = 15. 5

3.15. , " # - 3.3

+ ’ , ))

) .

	& AB	AC .
1.	A(1, −2, 3), B(0, −1, 2), C (3, −4, 5).
2.	A(0, −3, 6), B(−12, −3, −3), C (−9, −3, −6).
3.	A(3, 3, −1), B(5, 5, −2), C (4, 1, 1).
4.	A(−1, 2, −3), B(3, 4, −6), C (1, 1, −1).
5.	A(−4, −2, 0), B(−1, −2, 4), C (3, −2, 1).
6.	A(5, 3, −1), B(5, 2, 0), C (6, 4, −1).
7.	A(−3, −7, −5), B(0, −1, −2), C (2, 3, 0).
8.	A(2, −4, 6), B(0, −2, 4), C (6, −8, 10).
9.	A(0, 1, −2), B(3, 1, 2), C (4, 1, 1).

10.	A(3,	3,	−1), B(1,		5,	−2), C (4,		1,	1).
11.	A(2,	1,	−1), B(6,		−1,	−4), C (4,		2,	1).
12.	A(−1,	−2, 1),		B(−4, −2,			5), C (−8,		−2, 2).
13.	A(6,	2,	−3), B(6,		3,	−2), C (7,		3, −3).
14.	A(0,	0,	4), B(−3,		−6,	1), C (−5,		−10, −1).
15.	A(2,	−8,	−1), B(4,		−6,		0), C (−2,		−5, −1).
16.	A(3,	−6,	9), B(0,		−3,	6), C (9,		−12, 15).
17.	A(0,	2,	−4), B(8,		2,	2), C (6,		2,	4).
18.	A(3,	3,	−1), B(5,		1,	−2), C (4,		1,	1).
19.	A(−4,	3,	0),	B(0,	1,	3),	C (−2,	4,	−2).

129

20.	A(1,	−1,	0),	B(−2,		−1, 4),				C (8,		−1,			−1).
21.	A(7,	0,	2),	B(7,	1,		3),		C	(8,	−1,		2).
22.	A(2,	3,	2),	B(−1,		−3, −1),				C (−3,			−7,		−3).
23.	A(2,	2,	7),	B(0,	0,		6),		C (−2,			5,		7).
24.	A(−1,	2,	−3), B(0,			1, −2),				C (−3,			4,		−5).
25.	A(0,	3,	−6),	B(9,		3,	6),		C (12,			3,		3).
26.	A(3,	3,	−1),	B(5,		1, −2),				C (4,		1,		−3).
27.	A(−2,	1,	1),	B(2,		3, −2),				C (0,		0,		3).
28.	A(1,	4,	−1),	B(−2,		4, −5),				C (8,		4,		0).
29.	A(0,	1,	0),	B(0,	2,		1),		C (1,		2,	0).
30.	A(−4,	0,	4),	B(−1,		6,		7),		C (1,		10,		9).
31.	A(−2,	4,	−6), B(0,			2,		−4),			C (−6,			8,	−10).

3.16. $ %

) .

. 3.26. ( )

"
			a , b ,	c ,
: –
		a , – b , – c .
			'-
/ a ,	b ,	c

,

) . % -

						, -
	c , a							b -
	) .
						a	b c , -
	:
	1.	$				,
					c
		a	b .
	2.	c .

130