Все лекции по аналитический геометрии

$ a . , i , j , k - .

" : a = a1i + a2 j + a3k = (a1 , a2 , a3 ) 4 a1 , a2 , a3 a .

#$ (

1.	a + b = a i + a j + a k + b i + b j + b k = (a + b )i + (a + b ) j + (a + b )k = (a + b , a														2	+ b , a + b ) ;
	1	2	3	1	2	3	1	1	2	2	3	3	1	1		2	3	3
2.	λa = λ(a1i + a2 j + a3k ) = λa1i + λa2 j + λa3k = (λa1 , λa2 , λa3 ) .

" , ,

: 5 – OA = a = (a , a		, a )	A = a(a , a , a )
1	2	3	1	2	3

)* "

B" A(a1 , a2 , a3 ) B(b1 , b2 , b3 ) .

': AO + OB = AB

A O

AB = AO + OB = OB − OA = (b1 , b2 , b3 ) − (a1 , a2 , a3 ) = (b1 − a1 , b2 − a2 , b3 − a3 )

AB = (b1 − a1 , b2 − a2 , b3 − a3 )

% (

z

" a = OA = (a , a

, a ) , :

1

2

3

=

=

OA

=

OB

2

+

AB

2

=

OC

2

+

CB

2

+

BA

2

,

A

a

OA

=

=

=

=

,

OC

OC

a1i

a1

i

a1

=

=

=

=

,

CB

CB

a2 j

a2

j

a2

O

y

BA

=

BA

=

a3k

=

a3

k

=

a3

.

":

C B

x

&

" A(a1 , a2 , a3 )

( AB :

AB = (b1 − a1 , b2 − a2 , b3 − a3 ) .

: dist( A, B) = AB =

a = a12 + a22 + a32

* &" 4.

A

B(b1 , b2 , b3 ) .

B

		= (b		− a )2	+ (b − a )2		+ (b − a )2
	AB
		1		1	2	2	3	3

- * 2 .

§5.

$ " a b :

b

(a, b ) =

a

b

cosϕ ,

a

b

- , ϕ - .

ϕ

0 ≤ ϕ ≤ π 0o ≤ ϕ ≤ 180o .

a

11

(

(. " a = (a1 , a2 , a3 ) b = (b1 , b2 , b3 ) , ,

:

+ a b

+ a b .

(5.1)

(a, b ) = a b

1

1

2

2

3

3

&.

:

b

2

=

2

+

2

− 2

cos

ϕ

(Th.cos)

b − a

a

− b

a

b

a

b

ϕ

1

2

2

2

a

": (a, b ) =

(

a

+

b

−

a

− b

) =

2

=1 (a12 + a22 + a32 + b12 + b22 + b32 − (b1 − a1 )2 − (b2 − a2 )2 − (b3 − a3 )2 ) =

2

=1 (2a1b1 + 2a2b2 + 2a3b3 ) = a1b1 + a2b2 + a3b3 . 2

', ) (Th.cos) , ϕ = 0

ϕ = π ( ).

) (5.1) . ( .6 %. * (5.1) .

1)(a, a) = a 2 ≥ 0 ( ) - ;

2)(a, b ) = (b, a) - ;

3)λ(a, b ) = (λa, b ) = (a, λb ) , λ R - ; (a + b, c ) = (a, c ) + (b, c )

4)				- .
(a, b	+ c ) = (a, b ) + (a, c )

&.

1 2 ; 3 4 ) (5.1).

& 4- : (a, b + c ) = a1 (b1 + c1 ) + a2 (b2 + c2 ) + a3 (b3 + c3 ) =

=(a1b1 + a2b2 + a3b3 ) + (a1c1 + a2c2 + a3c3 ) = (a, b ) + (a, c ).

+

(a, b ) > 0	ϕ (0o , 90o ) ;
(a, b ) < 0	ϕ (90o ,180o ) ;
(a, b ) = 0	ϕ = 90o , . . a b .

%. 0

(a, 0) = (0, a) = 0 .

!$ "

a	u :
	u - ,
ϕ	a - ,
n	ϕ - a u .
4 a u :		pr a	=		a		cosϕ
		u

12

": ,

a u .

* . " n - u , :

=

(a, n)

pru a

n

cosϕ

(a, n)

a

n

&:

=

=

a

cosϕ = pru a.

n

n

%. / : pru a = prn a .

$

1.pru λa = λ pru a , λ .

2.pru (a + b ) = pru a + prub .

&.

" n u :

(λa, n)

a

n

cosϕ

1.

pru λa

=

=

λ

=

λ pru a

n

n

(a + b, n)

(a, n)

(b, n)

2.

pru (a

+ b ) =

=

+

= pru a

+ pru b.

n

n

n

.

pru (λ1 f1 + λ2 f2 + ... + λn fn ) = λ1 pru f1 + λ2 pru f2 + ... + λn pru fn .

":

z

i , j , k

- ,

Oxyz -

a = (a1 , a2 , a3 ) - , :

a

(a, i )

k

prOx a

=

= (a, i ) = a1

1 + a2

0 + a3

0

= a1 ,

i

prOy a = a2 ,

i

j

y

prOz a

= a3 .

x

: " !

.

, )*

4

& .

". " u - n , n = 1 , α , β , γ - u

Ox, Oy, Oz .

(: pr n	=		n		cosα = cosα , pr		n = cos β pr n = cos γ .
Ox						Oy	Oz
cosα , cos β cos γ						n .

)* (

1.	n = (cosα , cos β , cos γ ) - u ( . .		n		= 1 ).
2.	cos2 α + cos2 β + cos2 γ = 1 - - .

13

# * ( (

$ a, b , c, d.... , , Oxy .

(: a = (a1 , a2 , 0) = (a1 , a2 ) , b = (b1 , b2 ) , c = (c1 , c2 ) .

( .

, ) .

1: (a, b ) = a b + a b ,					a
							= a 2		+ a 2 .
1	1	2	2				1		2

( .

%:

b

ϕ

a

$ - :

1 a b .

ϕ . 1 .

																																	2			2													2				2
S		=				sin ϕ =															1 − cos2 ϕ =																− (					cosϕ)2 =
			a		b									a			b															a			b				a		b							a				b		− (a, b )2 =
= (a 2 + a 2 )(b 2											+ b 2 ) − (a b + a b )2																		= a 2b 2 + a									2b 2		+ a			2b	2 + a		2b 2		− a			2b 2			− a	2b 2	− 2a a b b =
		1		2			1				2							1			1					2	2			1				1			2	2		1			2	2		1				1	1			2	2	1		2	1	2
= (a b − a b )2											=		a b − a b											.
		1	2	2			1						1		2			2			1
':				S			=		a1b2 − a2b1
', a b , S																																											= 0 , a1b2 − a2b1 = 0 ,
. . a b = a							b												a1		=				b1
		1	2	2			1											a2							b2

§6.

5 – .

#

" a = (a1 , a2 ) b = (b1 , b2 ) - .

1 !, .

	a	a
!:	M = 1		2	.
	b1	b2

'. . ( ) :

det M =		M		=	a1	a2	= a b − a b .
					b1	b2	1	2	2	1

%. + ,

a b ( det M = S − ),

, “+” ,

, “– ” , – .

(: ( )

2 a b ( a b ), ( det M = 0 ). &: ( . - ) 5).

": : ,

! ( . . a b a1b2 = a2b1 ).

14

# "

" a = (a1 , a2 , a3 ) , b = (b1 , b2 , b3 ) c = (c1 , c2 , c3 ) - .

a

a

a

!

:

1

2

3

M = b1

b2

b3

.

c2

c3

c1

:

M = a, b, c

.

'. ' ! M = [a, b, c]

:

det M =

=

a1

a2

a3

= a

b2

b3

− a

b1

b3

+ a

b1

b2

( !)

M

b

b

b

1

2

3

1

c c

2

c c

3

c c

c1

c2

c3

2

3

1

3

1

2

%. %

. .

1 : det M = a1 (b2c3 − b3c2 ) − a2 (b1c3 − b3c1 ) + a3 (b1c2 − b2c1 ) = = a1b2c3 − a1b3c2 − a2b1c3 + a2b3c1 + a3b1c2 − a3b2c1 = a1b2c3 + a2b3c1 + a3b1c2 − a1b3c2 − a2b1c3 − a3b2c1 .

" :

"+ " "− "

1." ;

2.2 , 0;

3." ;

4. 2 ! - , ,

, ;

1: ' a, b, c + d

a, b, c a, b, d .

5. " ! , ,

( ! !

, !, ).

4 :

1) 1 det = a, b , c = − det a, c, b .

2) 1 det = [a, a, c ] = 0 ( D = det [a, a, c ] = − det [a, a, c ] D = 0 ).

3)1 det = λa, b, c = λ det a, b, c .

4)1 det a, b, c + d = det a, b, c + det a, b, d .

5)1 det a, b, c + λa = det a, b, c + λ det a, b , a = det a, b, c .

15

$

					a		a		a
"				=		1		2	3		- !.
	M = a, b, c					b	b		b
						1	2		3
					c1		c2		c3

! M T M !:

a

b

c

M

T

T

=

1

1

1

.

= a, b , c

a

b c

2

2

2

a3

b3

c3

". " ! - !, ! –

( ! ).

(: ' !

:

			T		a1	b1	c1
				=	a	b	c	= a b c + a b c + a b c − a b c − a b c − a b c = det M .
	det M T = a, b, c
					2	2	2	1 2 3 2 3 1 3 1 2 1 3 2 2 1 3 3 2 1
					a3	b3	c3
	:					« »
	«!».

§7.

" !

, , ,

( ).

# (

c

b

a

ν

b

ϕ

a

" a, b, c -

.

( ,

« »,

,

« ».

( :

" a, b - , ϕ -

.

': v

" a b , :

1.v = a b sin ϕ ,

2.v a b ,

3.a, b, v - , . . v

« ».

16

% :

a)		v		= S − ;
b)		v		= 0 a b	( 2 3 );
c)	v a b .
':									× b .
					v	= a, b	= ab	= a

1. [a, a] = 0 , a ;

2.

a, b

= − b, a

- ;

3.

λa, b

=

a, λb

= λ a, b , λ R - ;

+ b

, c

4.

a

= [a, c ] + b, c

+

[

]

- .

a, b

+ c

= a, b

a, c

', 3 4 .

&.

1)	S − = 0 , , , .
2)	" a b S − ,
.
3)	" λ , S − λ ,

( ) λ . :

λa, b		= a, λb = λ a, b .
4)	( . ).
":
	k			" i , j , k - .
				( - :
						= k ;
				i , j
			j
				j , k		= i ;
i				k , i		= j .

$ & a b

	b	i	j	k
a
i		0	k	− j
j		− k	0	i
k		j	− i	0

17

(

" a = (a1 , a2 , a3 ) b = (b1 , b2 , b3 ) . ( :

[a, b ] = [a i + a	2	j + a k , b i + b			j + b k ] =	( 3 4)
1		3	1	2	3

= a1 b1 0 + a1 b2 k + a1 b3 (− j ) + a2 b1 (−k ) + a2 b2 0 + a2 b3 i + a3 b1 j + a3 b2 (−i ) + a3 b3 0 =

=(a2b3 + a3b2 )i − (a1b3 + a3b1 ) j + (a1b2 + a2b1 )k =

i j k

= a1	a2	a3		" ) .
b1	b2	b3
					i	j		k
( :			[a, b ] =		a	a	2	a
					1			3
					b1	b2		b3

". &

.

!$ "

B

a

A

a′ B′

πA′

	c
a	b
	c ′
a′	b ′
π

c

a

a′

πv

: AB - ,

π - ,

A′, B′ - ! A B π , a′ = A′B′ - ! AB = a ,

:

a)$ !;

b)" ! ! ,

(a + b )′ = c′ = a′ + b′ .

" a c , c ≠ 0 .

$ π , c π . " a′ - ! a π , :

[a, c ] = [a′, c ]

&.

[a, c ] [a′, c ] ,

a, a′, c .

& [a, c ] [a′, c ] , . . ,

( c a′ ( . .)). ( : [a, c ] = [a′, c ] .

"

( ):
	a	+ b, c	= [a, c	] + b, c .

18

&. c

a a + b

b

1.2 c = 0 , .

2." c = 1 :

a	+ b, c	= (a + b )′, c		= a′ + b′, c			,		c		= 1 ,
π c
90 (
- c ,
), : [a′, c ]				+ b′, c = a′ + b′, c .

( : a + b, c = (a + b )′, c = a′ + b′, c = [a′, c ] + b′, c = [a, c ] + b, c .

3.

, c

- ,

c

≠ 0 , :

c

c

c

a

+ b, c

=

c

a

+ b,

=

c

a,

+ b,

= [a, c

]+ b

, c

. ( .

c

c

c

* .

§8.

c

" a, b, c -

d

.

b

$ " a, b c

:

(a, b, c )

≡ abc

≡ ([a, b ], c ) .

a

" ) :

" :

a = (a1 , a2 , a3 ) , b = (b1 , b2 , b3 ) , c = (c1 , c2 , c3 )

-

.

i

j

k

a a

a

2

a

a a

(

, b

] =

a

a

a

= i

3

− j

1

3

+ k

1

2

.

[a

2

1

3

b2

b3

b1

b3

b1

b2

b1

b2

b3

a2

a3

a1

a3

a1

a2

c1

c2

c3

a1

a2

a3

:

− c

+ c

=

=

.

a

a

a

b

b

b

([a, b ], c ) = c

1

b b

2

b b

3

b b

1

2

3

1

2

3

2

3

1

3

1

2

b1

b2

b3

c1

c2

c3

( ).

			a	a	a
( :			1	2	3		- a, b c
	(a, b, c ) =		b	b	b		-
			1	2	3		.
			c1	c2	c3

: - a, b c , ,

.

-

1.2 a, b c , -

,	a, b, c - (a, b, c ) = 0 .
&.

19

(a, b, c ) = 0		([a, b ], c ) = 0		[a, b ] c	[a, b ] = 0,
					c a, b;

	, b ,
a			a, b, c .
c	a, b;

2." a, b, c - ;

v	V - 73 , a, b, c .
				V , a, b, c − ,
		( (a, b , c ) =
				−V , a, b, c − .

c

b

θ

ϕ

a

% ‘+’ ,

a b ,

a, b, v ). / ‘ − ’ .

%:

(a, b, c )

= V

( ).

&.

=

c

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

, c )

a, b

cosϕ = (

a

b

sin α ) (

c

cosθ ) .

a

sin α = S

%

b

-

,

c

cosθ = ± h - ! c

v =

a, b

, h - .

c, ,

v , , a, b, c - (

: ' !, 3 ,

73 , .

$ 1. 2. . 5 ".

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

( - . &.

			b1	b2	b3		a1	a2	a3
(a,[b, c	]) = ([b, c	], a ) = (b , c, a ) =	c1	c2	c3	=	b1	b2	b3	= (a, b, c ) = ([a, b ], c ) .
			a1	a2	a3		c1	c2	c3

4." - " (

! ). 1: (a, b, c ) = − (b, a, c ) .

& .

5.- .

1: (α a + βb, c, d ) = α (a, c, d ) + β (b, c, d ) ( α β ).

&.

, . . ( )

(α a + βb, c, d ) = ([α a + βb , c], d ) = (α[a, c] + β[b, c], d ) =

= α ([a, c], d )+ β ([b, c], d ) = α (a, c, d )+ β (b, c, d ).

5 .

20