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Высшая математика. Том 1

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3.		,
3.	c	,	a	b
.

. 3.27.

a × b .

, -

1. & ,

, , , ,

a × b

sinϕ .

sin (ab)=

/ ,

b :

S . =

sinϕ .

b :

S - .

sinϕ .

+ ,

- ! - a

b :

λ (a × b) = (λ a)× b = a ×

(λb).

3. $ -

a ,

b ,

c :

a × (b + c) = a × b

+ a × c .

4. #

a × b

= −b × a .

. 3.28. &

, -.

131

* & " &

/ , , -

, ) -

. & a × a = 0 .

+ y

7 , a

= x i

j + z k

b = x i + y

j + z

k , -

a × b

= i

− j

+ k

# 1.

(2a

+ 3b)×

(a − 2b).

2a + 3b)×

− 2b) = 2a × a − 2a ×

3b × a

− 3b × 2b

' a × a = 0

a × b = −b × a

, :

(2a + 3b)× (a − 2b) = −4a

× b − 3a

× b

= −7a × b ,

7b × a .5

# 2. & , 2a + b

b , ,

= 1,

= 2 a;b

4 S - .

2a + b

sinϕ .

& (2a + b)× b = (2a)× b + b × b

= (2a)× b

' (λ a)× b = λ (a × b),

S - . =

(2a)× b

sinϕ =

sin (a;b)= 1 2

sin

= 2

= 1. 5

	#		3.

b = 3i	− j −	4k .
			i
	4a × b =		2
			3

&
	a = 2i	+ 3 j − k

3 −1

−1

3 −1 = i

−1

−4

− j

−4

+ k

−1

= −13i

+ 5 j − 11k .5

−1

−4

132

# 4. & ABC , A(2; 3; 1), B(–1; –2; 0),

C(–3; 0; 1).

= 1

4 S - .

AB × AC

;

AB = (−1 − 2;−2 − 3;−1) = (−3;−5;−1);

(−5;−3;0).

= (−3 − 2;−3;1 − 1) =

−5

−1

−3

−1

−3

AB × AC

−3 −5 −1

= i

−3

− j

−5

+ k

−5

= 3i − 5 j

+ 16k .

−5 −3 0

x2 + y2

+ z2 , S

32 + 52 + 162 =

AB × AC

- .

290 . 5

= (n, p, −1) ,

= (4 p, −q,2) ,

(8,13, −3).

c =

& n, p, q ,

, b

, )

4 ' a

, :

= −1 . 6

4 p

−q

−1

# & (

2 p = q)

& (2n = −4 p).

−q

4 p

)

, :

b c = 4 p 8 − q 13 − 2 3 = 0 32 p − 13q − 6 = 0 .

/ , , ’

n, p, q.

n = −2 p

n = −2,

n = −2 p

2 p

= q

= 1,

32 p = 16q

= 2.

32 p − 13q − 6 = 0

16q − 13q = 6

# 6. &

a =

µ1 p +

η1 q

= µ2 p

+ η2 q ,

µ1 = 6 ; µ2 = 1; η1 = −1; η2 = 2 ;

8 ; q = 1

; (pq)= π

4 # ,

b ,

)

S =

× b

a × b =

(µ1 p

+ η1 q)

× (µ2 p + η2 q)

= µ1µ2 p

× p

µ1η2 p × q

+ µ2η1 q × p

+ η1η2 q × q

133

& ) :

p × q = −q × p ( )
		= 0 .
p × p = q	× q	= 0 .

a × b = 1η2 p × q − 2η1 p × q = ( 1η2 − 2η1) p × q .

& , -

µ1η2 − µ2η1

sin (p, q).

2 − 1 (−1) 8 1

sinπ

= 13 4

= 26 3

. 5

3.17. , " # - 3.4

+ ’ , ))

) .

& a b .

1. a = p + 2q, b = 3p − q;

= 1,

= 2,

(p q) = π 6.

2. a = 3p + q, b = p − 2q;

= 4,

= 1,

(p q) = π 4.

3. a = p − 3q, b = p + 2q; p = 1 5, q = 1,

(p q) = π 2.

4. a = 3p − 2q, b = p + 5q; p = 4, q = 1 2, (p q) = 5π 6.

5. a = p − 2q, b = 2p + q;

= 2,

= 3,

(p q) = 3π 4.

6. a = p + 3q, b = p − 2q;

= 2,

= 3,

(p q) = π 3.

7. a = 2p − q, b = p + 3q;

= 3,

= 2,

(p q) = π 2.

8. a = 4p + q, b = p − q;

= 7,

= 2, (p q) = π 4.

9. a = p − 4q, b = 3p + q;

= 1,

= 2,

(p q) = π 6.

10.

a = p + 4q,

b = 2p − q;

= 7,

= 2,

(p q) = π 3.

11.

a = 3p + 2q,

b = p − q;

= 10,

= 1,

(p q) = π 2.

12.

a = 4p − q,

b = p + 2q;

= 5,

= 4,

(p q) = π 4.

13.

a = 2p + 3q,

b = p − 2q;

= 6,

= 7,

(p q) = π 3.

14.

a = 3p − q,

b = p + 2q;

= 3,

= 4,

(p q) = π 3.

15.

a = 2p + 3q,

b = p − 2q;

= 2,

= 3,

(p q) = π 4.

134

16.

a = 2p − 3q,

b = 3p + q;

= 4,

= 1,

(p q) = π 6.

17.

a = 5p + q,

b = p − 3q;

= 1,

= 2,

(p q) = π 3.

18.

a = 7p − 2q,

b = p + 3q;

p = 1 2,

q = 2, (p q) = π 2.

19.

a = 6p − q,

b = p + q;

= 3,

= 4,

(p q) = π 4.

20.

a = 10p + q,

b = 3p − 2q;

= 4,

= 1,

(p q) = π 6.

21.

a = 6p − q,

b = p + 2q;

p = 8,

q = 1 2,

(p q) = π 3.

22.

a = 3p + 4q,

b = q − p;

= 2,5,

= 2,

(p q) = π 2.

23.

a = 7p + q,

b = p − 3q;

= 3,

= 1,

(p q) = 3π 4.

24.

a = p + 3q,

b = 3p − q;

= 3,

= 5,

(p q) = 2π 3.

25.

a = 3p + q,

b = p − 3q;

= 7,

= 2,

(p q) = π 4.

26.

a = 5p − q,

b = p + q;

= 5,

= 3,

(p q) = 5π 6.

27.

a = 3p − 4q,

b = p + 3q;

= 2,

= 3,

(p q) = π 4.

28.

a = 6p − q,

b = 5q + p;

p = 1 2,

q = 4,

(p q) = 5π 6.

29.

a = 2p + 3q,

b = p − 2q;

= 2,

= 1,

(p q) = π 3.

30.

a = 2p − 3q,

b = 5p + q;

= 2,

= 3,

(p q) = π 2.

31.

a = 3p + 2q,

b = 2p − q;

= 4,

= 3,

(p q) = 3π 4.

3.18. 0 ' $ %

- a , b , c ,

, -

	a × b	c . # -

(abc) = (a × b) c . &, .

1. . . 7 -

’ ,

, , (abc) = ±V .
/ , ’ V =

					(abc)	,
		1


’ ( ) V	=		(abc)	.
’ ( ) V	=		(abc)	.
	6
	6

135

. 3.29. 8

a ,

b ,

(abc) > 0 ,

< 0 .

a , b , c – , (abc)

2. $ -

a ,

b ,

(abc)

= (a × b) c = a

(b × c)

3. # -

$ ,

(abc),

(bac)

= (b

× a)

c = −

(a × b)

c = − (abc).

, -

4. 7 (abc) = 0

a ,

b ,

c –

a = x i + y j + z k

b = x2 i

+ y2 j + z2 k

= x3 i

+ y3 j + z3 k , , ) -

, :

(abc)

/ , (abc)

, -

* " & " &

6 ,

" &

) .

( , ,

a , b ,

, (abc) ≠ 0 .

136


		# 1.									& ’ A(2;																						–2;	0),

B(–1; 4; –4), C(4; –8; 5), D(1; –7; 0). # AB ,

AC		AD ?						1


		4V				=					(AB AC AD)											.
		4V				=					(AB AC AD)											.
						6
						6						−3					6		−4
									=														= −1		6	−4	+ 5	−3		−4	= −6 − 35 = −41.

		(AB AC AD)												2			−6		5
													−1				−5		0						−6	5		2		5
													−1				−5		0
		V			=	1	−41						=		41			= 6	5		.
						1									41				5

						6										6			6
						6										6			6

		/ (AB AC AD) < 0 , . 5
		# 2.
		# 2.								&' a , b ,																								c ,

	a	= −i +		3 j +		2k , b								= 2i			− 3 j		− 4k , c						= −3i + j + k .
		4							−1							3	2			=			−1	3		0	= −5		−1	3		= 15 ≠ 0 , -


				(abc) =					2					−3 −4									2	−3		0
									−3							1	1						−3	1		−5			2	−3
									−3							1	1						−3	1		−5

a , b ,			c – . 5

3.19. , " # - 3.5

+ ’ , ))

) .

&' a , b c?

1. a = {2, 3,	1},	b = {−1, 0, −1}, c = {2, 2, 2}.
2. a = {3, 2,	1},	b = {2, 3, 4}, c = {3, 1, −1}.

3. a = {1, 5, 2}, b = {−1, 1, −1}, c = {1, 1, 1}. 4. a = {1, −1, −3}, b = {3, 2, 1}, c = {2, 3, 4}. 5. a = {3, 3, 1}, b = {1, −2, 1}, c = {1, 1, 1}.

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

8. a = {4, 3, 1}, b = {6, 7, 4}, c = {2, 0, −1}. 9. a = {3, 2, 1}, b = {1, −3, −7}, c = {1, 2, 3}. 10. a = {3, 7, 2}, b = {−2, 0, −1}, c = {2, 2, 1}.

137

11.

a = {1,

−2,

6} ,

b = {1,

0, 1}

c = {2,

−6,

17} .

12.

a = {6,

4} ,

b = {−1,

−2,

−1} ,

c = {2,

2} .

13.

a = {7,

4} ,

b = {−1,

−2,

−1} ,

c = {4,

4} .

14.

a = {2,

2} ,

b = {4,

5} ,

c = {2,

−1} .

15.

a = {5,

4} ,

b = {−1,

0, −1} ,

c = {4,

4} .

16.

a = {3,

10,

5} ,

b = {−2,

−2,

−3}

c = {2,

4, 3}.

17.

a = {−2,

−4, −3} ,

b = {4,

1} ,

c = {6,

4} .

18.

a = {3,

−1} ,

b = {1,

−1} ,

c = {8,

−2} .

19.

a = {4,

2} ,

b = {−3,

−3,

−3} ,

c = {2,

2} .

20.

a = {4,

2} ,

b = {9,

5} ,

c = {1,

1, −1}.

21.

a = {5,

4} ,

b = {4,

3} ,

c = {9,

8} .

22.

a = {3,

2} ,

b = {1,

0} ,

c = {8,

11,

6} .

23.

a = {4,

−1,

−6} ,

b = {1,

−3,

−7} ,

c = {2,

−1, −4} .

24.

a = {3,

0} ,

b = {−5,

−4, −5} ,

c = {4,

4} .

25.

a = {3,

3} ,

b = {8,

6} ,

c = {1,

−1} .

26.

a = {1,

−1,

4} ,

b = {1,

c = {1,

−3,

8} .

27.

a = {6,

4} ,

b = {−1,

−2,

−1} ,

c = {2,

2} .

28.

a = {4,

1} ,

b = {−9,

−4, −9} ,

c = {6,

6} .

29.

a = {−3,

3} ,

b = {−4,

6} ,

c = {3,

−1} .

30.

a = {−7,

10, −5} ,

b = {0,

−2,

−1} ,

c = {−2,

4, −1} .

31.

a = {7,

6} ,

b = {2,

1} ,

c = {19,

11,

17} .

3.20.( )

( ) )) . /

), -

, .

" A1, A2, A3, A4. # -
A4 A1A2A3.
& A1 3		) :
		= ( x2 − x1; y2 − y1; z2 − z1 ) ,
	A1 A2
			− y1; z4 − z1 )
A1 A3 = ( x3	− x1; y3 − y1; z3 − z1 ) , A1 A4 = ( x4 − x1; y4

138

	h
	A3
A	A2
1

. 3.30. # A1, A2, A3, A4

& ) ) , ` :

h .

h =

3VA A

A A

1 2

A A

SA A

& , ’

(A A A A A A )

A1A2A3A4

1 2 1 3 1 4

(A1 A2 A1 A3 A1 A4 )

, -

x2 − x1

y2 − y1

z2 − z1

(A1 A2 A1 A3 A1 A4 ) =

x3 − x1

y3 − y1

z3 − z1

x4 − x1

y4 − y1

z4 − z1

# ( ) -

(A A

× A A ).

A A

1 2

( (

A1 A2

× A1 A3 ) :

− x1

y2 − y1

z2 − z1

(A1 A2

× A1 A3 ) =

x3 − x1

y3 − y1

z3 − z1

', :

(A1 A2

A1 A3 A1 A4 )

(A A A A A A )

1 2 1 3 1 4

h =

A1A2A3A4

SA1A2A3

(A1 A2

× A1 A3 )

(A1 A2

× A1 A3 )

139

' ’ A1, A2,

A3, A4

, A4 A1A2A3,

A1(1;5;–7),

A2(–3;6;3), A3(–2;7;3),

A4(–4;8;–12).

4 & :

= (−3 − 1; 6 − 5; 3 + 7) = (−4;1;10) ,

A1 A2

= (−2 − 1; 7 − 5; 3 + 7) = (−3; 2;10) ,

A1 A3

(−4 − 1; 8 − 5; − 12 + 7)

= (−5; 3; − 5) .

A1 A4

& (A1 A2 A1 A3 A1 A4 ) =

−4 1 10

−14 7 0

0 7 0

−3

= −7

= 105.

−5 3 −5

1 3 −5

−5

'’ :

(A A A A A A )

105 = 17,5.

A A

& (A1 A2 × A1 A3 ) =

1 10

−4 10

−4 1

−4 1 10

= i

− j

−3

+ k

−3

= −10i +

10 j − 5k

−3

( ) :

+ y2 + z2

(−10)2 + 102 + (−5)2 =

225 = 15 .

A A

× A A

h =

3VA A

(A1 A2

A1 A3

A1 A4 )

105

= 7 .5

SA A

(A A

× A A )

3.21. , " # - 3.6

+ ’ , ))

) .

	' ’ A1, A2, A3, A4 -
, A4 A1A2A3 .
1.	A1 (1, 3, 6), A2 (2, 2, 1), A3	(−1, 0, 1), A4	(−4, 6, −3).
2.	A1 (−4, 2, 6), A2 (2, −3, 0),	A3 (−10, 5, 8),	A4 (−5, 2, −4).

140

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