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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 ax+ by + cz  + 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, zabe 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 points p = 1,1,1 with a normal vector A  = 2,3,4

1. 

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

π: 2x + 3y + 4z + d = 0

2. 

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

π:

2x + 3y + 4z + d 

=

0

21 + 31+ 41 + d 

=

0

d

=

9

3. 

The equation of the plane is 2x + 3y + 4z  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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