, 1 2 (, , z) .
, ..
() = 0
:
,
.
, . (, , z), , .
:
.
. 0(0, 0, z 0), , 0 (A, B, C) :
A (x x 0) + B (y y 0) + C (z z 0) = 0.
. (, , z), , . .. - , , , , .
× = 0
,
.
.
+ + z + D = 0 (-D)
,
, :
a, b, c , , z.
.
- - (, , z),
- , , , .
a, b g - , , , z.
p .
:
xcosa + ycosb + zcosg - p = 0.
.
0(0, 0, z0) ++z+D=0 :
. , , (4; -3; 12) , .
, A = 4/13; B = -3/13; C = 12/13, :
A(x x0) + B(y y0) + C(z z0) = 0.
. , P(2; 0; -1)
Q(1; -1; 3) 3 + 2 z + 5 = 0.
3 + 2 z + 5 = 0 .
:
. , (2, -1, 4)
(3, 2, -1) + + 2 z 3 = 0.
: A x + B y + C z + D = 0, (A, B, C). (1, 3, -5) . , (1, 1, 2). .. , ,
|
|
, (11, -7, -2). .. , , .. 11×2 + 7×1 - 2×4 + D = 0; D = -21.
, : 11 x - 7 y 2 z 21 = 0.
. , , (4, -3, 12) , .
= (4, -3, 12). : 4 x 3 y + 12 z + D = 0. D :
16 + 9 + 144 + D = 0
D = -169
, : 4 x 3 y + 12 z 169 = 0
. 1(1; 0; 3), A2(2; -1; 3), A3(2; 1; 1),
A4(1; 2; 5).
1) 12.
2) 12 14.
3) 14 123.
123 .
= (2-1; 1-0; 1-3) = (1; 1; -2);
.
-4 4 = -8.
g g = 900 - b.
4) 123.
5) .
(3).
6) 123.
, .
2x + 2y + 2z 8 = 0
x + y + z 4 = 0;
.
.
, - . .
. y = f(x) , .
, , t.
. .
.
.
+ + = 0,
, , .. 2 + 2 ¹ 0. .
, :
- C = 0, ¹ 0, ¹ 0
- = 0, ¹ 0, ¹ 0 { By + C = 0}-
- = 0, ¹ 0, ¹ 0 { Ax + C = 0}
- = = 0, ¹ 0
- = = 0, ¹ 0
.
|
|
.
. (, ) , + + = 0.
. , (1, 2) (3, -1).
= 3 = -1 : 3 + = 0. .
: 3 2 + C = 0, = -1.
: : 3 1 = 0.
, .
M1(x1, y1, z1) M2(x2, y2, z2), , :
- , .
:
1 ¹ 2 = 1, 1 = 2.
= k .
. , (1, 2) (3, 4).
, :
.
+ + = 0 :
, k.
.
, .
. (a1, a2), a1 + a2 = 0
+ + = 0.
. (1, -1) (1, 2).
: Ax + By + C = 0. , :
1×A + (-1)×B = 0, .. = .
: Ax + Ay + C = 0, x + y + C/A = 0.
= 1, = 2 /A = -3, .. :
+ - 3 = 0
.
+ + = 0 ¹ 0, , , :
,
, , b .
. + 1 = 0. .
= 1, , = -1, b = 1.
.
+ + = 0 , ,
xcosj + ysinj - p = 0
.
, m× < 0.
, , j - , .
. 12 5 65 = 0. .
:
: ( 5)
:
; cosj = 12/13; sinj = -5/13; p = 5.
C , , , , .
. . , , 8 2.
|
|
: , a = b = 1; ab/2 = 8; a = 4; -4.
a = -4 .
: + 4 = 0.
. , (-2, -3) .
: , 1 = 1 = 0; x2 = -2; y2 = -3.
.
. y = k1x + b1, y = k2x + b2,
.
, k1 = k2.
, k1 = -1/k2.
. + + = 0 1 + 1 + 1 = 0 , 1 = l , 1 = l . 1 = l , .
.
,
.
. , 1(1, 1) = kx + b :
.
. (0, 0), + + =0
.
. 1(1, 1) , . 1:
(1)
x1 1 :
, 0 .
:
A(x x0) + B(y y0) + Ax0 + By0 + C = 0,
, , :
(1), :
.
.
. : y = -3x + 7; y = 2x + 1.
k1 = -3; k2 = 2 tgj = ; j = p/4.
. , 3 5 + 7 = 0 10 + 6 3 = 0 .
: k1 = 3/5, k2 = -5/3, k1k2 = -1, , .
. (0; 1), B(6; 5), C(12; -1). , .
: ; 4x = 6y 6;
2x 3y + 3 = 0;
: Ax + By + C = 0 y = kx + b.
k = . y = . .. , : b = 17. : .
: 3x + 2y 34 = 0.
.