
Coordinate
Geometry (Analytic Geometry) in Threedimensional Space 


Plane
in a threedimensional (3D) coordinate system 
Equations of
a plane in a coordinate space

The equation of a
plane in a 3D coordinate system 
The Hessian normal form of the
equation of a plane

The intercept
form of the equation of a plane 





Equations of a plane in a coordinate space

The equation of a
plane in
a
3D coordinate system 
A plane in space is defined by three points (which don’t all lie on the same line) or by a point and a normal
vector to the plane. 
Then, the scalar product of the vector
P_{1}P
= r 
r_{1}, drawn from the
given point P_{1}(x_{1},
y_{1}, z_{1}) of the plane to any point
P(x,
y, z)
of the plane, and the normal vector N
= Ai + Bj
+ Ck, is zero, that is 


which is called the vector equation of a
plane. 

Or,




Therefore, 
(x_{
} 
x_{1})
· A
+ (y_{ }
y_{1})
· B
+ (z_{ }
z_{1})
· C
= 0 
giving 
A
· x
+ B · y
+ C · z
+ D = 0 


where, D
= _{
}
(Ax_{1}+
By_{1}+
Cz_{1}) 
the general equation of a plane in 3D
space. 
If plane passes through the origin
O
of a coordinate system then its coordinates, x =
0, y
= 0, and z
= 0 plugged into the equation of the plane, give 
A
· 0
+ B · 0
+ C · 0
+ D = 0 => D = 0. 
Thus, the condition that plane passes through the
origin is D
= 0. 




The Hessian normal form of the
equation of a plane

Position of a plane in space can also be defined by the length of the normal, drawn from the origin to the plane
and by angles,
a, b and
g, that the normal forms with the
coordinate axes.

If
N°
is the unit vector of the normal then its components are, cosa,
cosb
and
cosg.

Then the projection of the position vector
r, of any point
P(x,
y, z) of the plane, onto the normal has the length
p. 
Since the projection is determined by expression 

that is,
r
· N°
= p or
written in the coordinates 

x
· cosa
+ y · cosb
+ z · cosg

p = 0 


represents the
equation of a plane in the Hessian
normal
form. 




The intercept form of the equation of a
plane

If,
l,
m
and n
are the intercepts of
x,
y
and z
axes and a plane respectively, then projections of these segments in
direction of the normal drawn from the origin to the
plane are all equal to the length of the normal, that is


By plugging these values of cosines into Hessian normal
form of the equation of plane, obtained is


the intercept form of the equation of plane. 












Coordinate
geometry contents 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 