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Equation of a Plane - 3 Points

 

Main Concept

A plane can be defined by four 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 is ax+ by + cz  + d  = 0, where a,b,c is the vector normal to the plane.

How to find the equation of a plane using three non-collinear points

Three points (A,B,C) can define two distinct vectors AB and AC. Since the two vectors lie on the plane, their cross product can be used as a normal to the plane.

1. 

Determine the vectors

2. 

Find the cross product of the two vectors

3. 

Substitute one point into the Cartesian equation to solve for d.

Example

Find the equation of the plane that passes through the points A = 1,1,1, B =1,1,0, C  = 2,0,3.

1. 

Determine the vectors

AB = 

xB xAi +yB  yAj +zB  zAk

11i +11j +01k

2i k

AC =

xC xAi +yC  yAj +zC  zAk

21i +01j +31k

i j + 2k

2. 

Determine the normal vector

AB x AC = ijk201112

AB x AC = i+3j +2k

3. 

The equation of the plane is

x + 3y + 2 z  + d  = 0

4. 

Plug in any point to find the value of d

 

d=

x + 3y + 2 z  

1 + 3 1 + 2 1 

d=

4

5. 

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

 

Change the three points on the plane and see how it affects the plane.

Point A

Point B

xA =

xB=

yA=

yB=

zA=

zB=

Point C

xC  = 

yC  =

zC =

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