n x 1, x 2, xi, , xn. m n xi, ,
(2.1)
ij , i , j . b i - .
A∙X = B, (2.2)
, , . , (A | B)
(2.3)
, , , ,
(A | B) = .
bi , , . , ( , , , ) ( ). .
.
n n :
.(2.5)
¹0, (2.4) , . = 0, (1) , .
, ¹0.
.
¹0, , , (2.4)
(2.6)
, ,, .
(i =1, 2,, n) , , ..
.
. . .
.
.
, :
.
.
.
:
,
.
. (2.4) ¹0. ,
( | B)=
, (2.4) ,
.. , , : .
|
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, . :
1) (.. );
2) , 0 (.. , 0);
3) , , 0 (.. , , 0).
. . .
. . , . .
.
.
1- . 1- 2.
.
2- . 1- . 2- 1-: 1- (-5), 2- , , 2- , :
3- 1-: 1- (-3) 3- ,
.
2- 1- .
3- . 11.
.
4- . 2- . 1- 2-: 2- 2 1- ,
3- 2-: 2- (-14) 3- ,
.
3- 4- 1- , 2- .
5- . 3-
.
6- . 3- . 1- 3-: 3- 1- , ࠠ
2- 3-: 3- 2- ,
.
5- 6- 3- . , :
. .
.
1 , .. 5, 1 , .. 2 . 1 , .. 3, 5 젠
_______________________________ ___________________
28, 22 . 2 3
3 = 4, 3 2
, 2 = -2,
|
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2 3 1
.
.
. , :
,
.
. .
:
, ,
, (, ),
, .
. , , ( 3.1), . . . . = .
: , ; a + b = c (.3.2). (. 3.3). ( 3.4)
. 3.2. . 3.3. |
. 3.1 |
λ : , , ( 3.5).
çλ ç = (3.1)
. 3.4 |
. 3.5. |
a - b = a + (-1) b.
pr AB = x 2 x 1= ç AB ç cos (α), (3.2)
α - (. 3.6). 1 1 . OY OZ:
pr AB = y 2 y 1= ç AB ç cos (β) (3.3)
prZAB = z 2 z 1= ç AB ç cos (γ), (3.3)
β γ OY OZ. 2 2 3 3 OY OZ .
cos (α), cos (β) cos (γ)
cos 2(α)+ cos 2(β)+ cos 2(γ) = 1. (3.4)
. 3.7. , . |
. 3.6. |
i, j k - , OY OZ ( ),
A1B 1 = i (x 2 x 1), A 2 B 2 = j (y 2 y 1) A 3 B 3 = k (z 2 z 1). (3.5)
(. 3.7):
AB = A 1 B 1 + A 2 B 2 + A 3 B 3 = i (x 2 x 1) + j (y 2 y 1) + k (z 2 z 1) ≡
≡{(x 2 x 1), (y 2 y 1), (z 2 z 1)}.
AB = i (x 2 x 1) + j (y 2 y 1) + k (z 2 z 1) .
AB = {(x 2 x 1), (y 2 y 1), (z 2 z 1)} (3.6)
|
|
. (x 2 x 1), (y 2 y 1) (z 2 z 1)
a, b
a = { x 1, y 1, z 1)} b = { x 2, y 2, z 2} :
a + b ={ (x 1+ x 2), (y 1 + y 2), (z 1 + z 2)}; (3.7)
λ a = {λ x 1, λ y 1, λ z 1}. (3.8)
a = { x 1, y 1, z 1)}; b = { x 2, y 2, z 2}
× b = (a,b) = ç a ç ç b çcos(Ð a, b) = x 1 x 2 + y 1 y 2 + z 1 z 2. (3.9)
: .
: × b = b × . , .
× a = x 1 x 1+ y 1 y 1 + z 1 z 1 = x 12+ y 12 + z 12 Þç a ç= (3.10)
. φ ,
1(1, -2, -3), 2(-3, 1, 1), 3(3, 2, 2).
. cosφ - . () cosφ. 2 1, . , 3 1.
= (-3-1, 1+2, 1+3) =(-4, 3, 4), = (3-1, 2+2, 2+3) = (2, 4, 5),
. → φ = 87045'54".
a b
c = a ´ b, :
1. a b,
2. ( )
ç c ç = ç a ç ç b ç sin (Ð a, b) = ç a ç ç b ç sin (φ), (3.11)
3. c : , c (. 3.8).
. 3.8.
: .
, - i, j, k
k = i ´ j, i = j ´ k, j = k ´ i. (3.12)
: a ´ b = - b ´ a. ( ) a ´ b = 0, (a çç b).
= a ´ b = = i (y 1 z 2 y 2 z 1) - j (x 1 z 2 z 1 x 2) + k (x 1 y 2 x 2 y 1). (3.13)
, , :
S = ç a çç b ç sin (φ).
a, b c -
a ´ b × c = a × b ´ c º abc = , (3.14)
. . ( ) , . . , ( ). , a, b, c.
|
|
V = ê abc ê; V = ê abc ê. (3.15)
. 1(1, -2, -3), 2(-3, 1, 1), 3(4, 3, -1), 4(3, 2, 2). 1 2 3 .
. 1 2 3 ,
,
- .
,
2.
,
,
, 3.