Векторная алгебра

" -"

"

, ,. ( /

:

"

*

" ,

-

",

*,

*

(

. "

* ",

*

*, *

"

(. #

* ,

. - " ,

,

* .

.

* " -

. ' /

"

* * ( .

'

- "

*

.

'

*

"

*- " ,

*

* ( "

* ,

- . ' "

*

* -

* ( "

.

1.

1.

! "

(#$)

,

", * , *

"

" .

1 * ,

,. ": *,

+ , + , * . , * .

2 - ", " ", ( .

* ,

!

! "

(!$).

1 * , : , ,. , *

.

+.

1 *

, * (

* ( *

*) * * .

2.

$ *

AB

" " & " " '. " ,

AB

a

r

= AB * ,,

( . 2.1). 2 " & ' a

AA = 0

% !&$ !$. # "

a

(

"

&

'

AB )

($%$) / "

AB

a

.

! a ! ! % b ( "

r

a = b ),

* , , ,

. .

r

r

=

r

,

b

a ↑↑ b

a

r

a = b .

* , " $& ( "$ $' % #

(

#"$ ! &$ ) ! !, $)* !&

$% +

.

1.1. & , !$

1.

! a "

λ . -(!$

! a

r

r

# ! + " #λ b = λ a

= a λ

, "

)

r

=

λ

r

;

b

a

)

r

,

λ > 0 ,

b ↑↑ a

)

r

,

λ < 0 .

b ↑↓ a

* * *

r

|| b ,

" * , " a

r

.$ #%* # !% " # α ≠ 0 , " a = αb . )*

r

, α ≠ 0 , a

b .

: a = αb

0

r

λ = 0 ;

r

, " λ a = 0 ,

(− 1) a

* *

a " − a .

2. ) a + b a b ( .

2.4, )

c ,

* " a

b * , "

b

* a ( . 2.4, ).

)

(

*

*

$$ . % / " a

b *

" * ( *

( . 2.4, ). %

*

,

.

" *

a

b

* * ,

- * ,

r

a + b .

) ( a1, a2 , a3 ,..., am

* ! %

$% +.

% / " * ,.

r

− b

3.

( # +)

a b c , "

a

r

r

r

r

− b

a

b + c

= a

. ' c

= a

!

*

( ( *.

1.

, . ( * . 5

*

" V1 .

2.

, . ( *, ,

*. 4

" V2 .

3.

( (

*

*

. 2 "

V3 .

4.

'

,

V

,

. (

(

* ,. (

* (

L1

L2 ,

*

r

r

r

≠ 0, b ≠ 0 .

, * a

+ b V a L1, b L2

, a

1.2. -, ! #+

2 * ( ' + !,

* *

#+).

! *

" .

1 * ,

* * * ,

, + &$ !$.

! * l " M (M l).

1. -, " M l

" M ′ ,

" AB l . " ( A′ B′ ( . 2.9).

2. $" #

,

! AB

l

A′B′ . (' ( " , * ,

* # # ! )* ! .)

3. " #

,

! AB

l

",

A′B′

, * A′B′

* * l ,

−

A′B′

* ".

& " * AB l " !r AB .

l

1 , " " , * , !r AB AB

l

l * ", :

!r AB =

AB

cosα , α -

AB , l .

l

! %$ $' % !$ #+)

*

- %

A′B′

,

0 ≤ α < π 2

!r

AB =

AB

cosα =

l

−

A′B′

,

π 2 < α ≤ π .

, ,. #!

# ! , :

1. !r

r

r

± !rb .

(a

± b )= !r a

l

l

l

α > 0

r

r

cosϕ = α

r

cosϕ

r

!l (αa ) =

αa

a

= α!l a

,

α < 0

r

r

cos(π − ϕ ) =

α

r

cos(π − ϕ ).

!l (αa ) =

αa

a

9

r

r

(− cosϕ ) = α

r

r

!l (αa ) = −α

a

a

cosϕ = α !l a .

1.

$,

! !

r

r

r

a1

, a2

,..., an

*

r

r

r

,

b = λ1a1 + λ2a2

+ ... + λn an

r

r

r

, ( ! #$&$ ,

2. ' a1

, a2

,..., an

r

r

r

λ1a1 + λ2 a2

+ ... + λn an = 0 .

(1)

'

r

r

r

,

( ! #$&$ ,

a1, a2 ,..., an

(1) * ,

" λ1, λ2 ,..., λn

,.

0 ! '

1.

1 (

"

%, , (

r

r

a1

b (a, b ≠ 0)