# 第二章 账户与借贷记账法 by dffhrtcv3

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```									3.7 Planar Equations
高等代数与解析几何                                3.7 平面方程

3.7.1 Planar equations decided by a point in plane
and azimuthal vector
 
Given a point M 0 and two vectors     a , b that are not collinear in
space, then the plane  via this point that parallel with vectors
                                        
a, b   is uniquely confirmed, vectors a , b is called azimuthal
vector of the plane.

Take affine coordinate system {O; e1 , e2 , e3} in space, and let
vector radius of point M 0 is OM 0  r0，the vector radius of an
arbitrary point M in plan  is OM  r . By coplanar condition,
                                        
MoM  u a v b    can be written as r  r0  u a  v b .
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高等代数与解析几何                                                  3.7 平面方程

The planar equation above is called vector form parametric
equation of plane  .

Let the coordinates of points M 0 , M are                     ( x0 , y0 , z0 ),( x, y, z),
                     
respectively, then r0  ( x0 , y0 , z0 ), r  ( x, y, z ) . Let
                     
a  ( X 1 , Y1 , Z1 ), b  ( X 2 , Y2 , Z 2 )
then
x  x0  X 1u  X 2 v
y  y0  Y1u  Y2 v
z  z0  Z1u  Z 2 v

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高等代数与解析几何                                                     3.7 平面方程

The equations above are called coordinate form arametric
equations of plan            , where u, v are parameters.
Point place form equation of plane.

x  x0      y  y0      z  z0
X1         Y1          Z1     0
X2          Y2          Z2

Example 3.32 There are three points
M1 ( x1 , y1 , z1 ), M 2 ( x2 , y2 , z2 ), M 3 ( x3 , y3 , z3 )

that are not collinear, find the equation of plane  via points
M1, M 2 , M 3.
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高等代数与解析几何                                        3.7 平面方程

Solution By vector form parametric equation, remove
           
parameters u, v, respectively, we obtain ( r  r1 , r2  r1 , r3  r1 )  0.

x  x1   y  y1     z  z1
x2  x1   y2  y1    z2  z1  0
x3  x1   y3  y1    z3  z1

can be rewritten as

x    y     z 1
x1   y1    z1 1
0
x2   y2    z2 1
x3   y3    z3 1
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高等代数与解析几何                                            3.7 平面方程

The above two equations are both called three points form
equation of plane.
Specially, if three known points are intersection points of
plane and three axes
M 1 (a, 0, 0), M 2 (0, b, 0), M 3 (0, 0, c)(abc  0),

then we obtain intercept form equation of plane

x y z
   1.
a b c

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高等代数与解析几何                                       3.7 平面方程

3.7.2 General equation of plane
Since any plane can be confirmed by a point M 0 ( x0 , y0 , z0 ) in it
              
and its azimuthal vector a ( X 1 , Y1 , Z1 ), b  ( X 2 , Y2 , Z 2 ) by point
,
direct form it is written as

Ax  By  Cz  D  0                                (*)
where

Y1     Z1      Z1     X1      X1 Y
A           ,B            ,C 
Y2     Z2      Z2     X2      X 2 Y2

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高等代数与解析几何                            3.7 平面方程

Equation（*）is called general form equation of a plane.
Theorem 3.4 Any planar equation in space can be represent-
ed as a linear equation on variables x, y, z , Conversely,
linear equation on each every variable x, y, z represents a
plane.

Let’s discuss several special cases of equation（*），i.e.
while some coefficients or constants of it equal to zeroes, plane
have some special positions for coordinate system.

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高等代数与解析几何                                 3.7 平面方程

1. If D  0 , (*) is Ax  By  Cz  0 origin (0, 0, 0)satisfies
,
the equation at this time, hence plane is via the origin.
Conversely, if the plane is via the origin, then D  0 obviously.

2. One of A, B, C is zero. On the condition of        D  0, if

A  0 , plane is parallel with axis   x , if B  0, the plane is
parallel with axis y . On the condition of C  0 , the plane is
parallel with axis z .

3. Two of A, B, C are zeroes, We have following conclusions.
Plane is parallel with coordinate plane yoz if and only if D  0,
B  C  0.The plane is just coordinate plane yoz if and only if
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高等代数与解析几何                               3.7 平面方程

D  0, B  C  0. Similarly for other cases.

3.7.3 Normal equation of plane

Given a fixed point M 0 and a vector n that is not equal to
zero, then a plane via the point M 0 is uniquely confirmed,

which is vertical with the vector n , and the vector is called
normal vector of the plane.

Under rectangular coordinates system in space       {o; i, j, k} ,
let vector radius of point M 0 be OM 0  r0 , and the vector radius
of an arbitrary point M in plane      be OM  r . The point M
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高等代数与解析几何                                      3.7 平面方程

   
is in the plane  if and only if the vector M 0 M  r  r0 is

vertical with the vector n , i.e.

A( x  x0 )  B( y  y0 )  C ( z  z0 )  0

Let D  ( Ax0  By0  Cz0 ), then the equation above is

Ax  By  Cz  D  0
which is called point normal form equation of plane.
Take the origin O as a point M 0 in plan specially, foot of a
perpendicular line that constructed toward plane  is p ，the
0
normal vector of  is identity vector n . If the plane is not via

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高等代数与解析几何                                         3.7 平面方程

0                
the origin, the positive direction of n and the vector op have
the same direction, while the plane is via the origin, take either
of the two directions that is perpendicular with the plane as the
0
positive direction of n . Let OP  p , then the normal
equation of vector is
0 
n r p  0
0
Let r  ( x, y, z), n  (cos 1,cos 2 ,cos 3 ) ，then

x cos 1  y cos  2  z cos 3  p  0
which is called coordinate form normal equation of plane.

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高等代数与解析几何                                       3.7 平面方程

Two characters of normal equation of plane.
1. The coefficient of linear factor is component of identity
vector, whose sum of squares is equal to 1；
2. The constant is not more than zero.
According to the two characters of normal equation of plane,
we can easily transform the general form equation of plane into
following form
 
n r  D  0.
Multiply both side of the above equation by
1                  1
k        

|n|          A2  B 2  C 2

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高等代数与解析几何                              3.7 平面方程

we can obtain normal equation of plane, hence    k   is
called normal factor.

Example 3.33 Given two points M 1 (1, 2,3), M 2 (3, 0  1)，
find the equation of perpendicular bisector plane  of line
segment M 1M 2 .

Answer             x  y  2z  1  0

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The End of Section 3.7

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