Equation of a Plane - Point and a Normal - Maple Help
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Home : Support : Online Help : Math Apps : Algebra and Geometry : Points, Lines, and Planes : Equation of a Plane - Point and a Normal

 

Equation of a Plane — Point and a Normal

 

Main Concept

A plane can be defined by five different methods:

• 

A line and a point not on the line

• 

Three non-collinear points (three points not on a line)

• 

A point and a normal vector

• 

Two intersecting lines

• 

Two parallel and non-coincident lines

 

The Cartesian equation of a plane π is a⋅x+ b⋅y + c⋅z  + d  = 0, where a,b,c is the vector normal to the plane.

How to find the equation of the plane through a point with a given normal vector

Let P xp,  yp,  zp be the point and A xa,ya, za be the normal vector.

1. 

Substitute xa,ya, za  into a, b, c respectively.

2. 

Plug in point P and solve for the last unknown variable d.

Example:

Find the equation of the plane that passes through the point p = 1,1,1 with a normal vector A⇀  = 2,3,4

1. 

Substitute  xa, ya,  za into a,  b,  c respectively

π: 2⋅x + 3⋅y + 4⋅z + d = 0

2. 

Plug in point P and solve for the last unknown variable d.

π:

2⋅x + 3⋅y + 4⋅z + d 

=

0

2⋅1 + 3⋅1+ 4⋅1 + d 

=

0

d

=

−9

3. 

The equation of the plane is 2⋅x + 3⋅y + 4⋅z −9 =0

 

Change the point and the normal see how it affects the plane.

Point A

Normal

x =

x=

y=

y=

z=

z=

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