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Equation of a Plane - 3 Points

 

Interact with this Math App in the MapleCloud! 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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