Three Points Determine a Plane

x

1

y

1

z

1

{1,2,3}

x

2

y

2

z

2

{3,1,2}

x

3

y

3

z

3

{2,3,1}

randomize

equation of plane:

x



3

+

y



3

+

z



3

2

3

with normal vector {2,2,2}

Three noncollinear points in three dimensions determine a unique plane with an equation of the form

Ax+By+Cz=D

, where

A

2

+B

2

+C

2

1

and

D

is the positive distance of the plane from the origin. The vector

(A,B,C)

is normal (perpendicular) to the plane and has norm (length) equal to 1.

For such an equation, the signed distance from a point

(x,y,z)

to the plane is given by

(A,B,C,D)·(x,y,z,-1)

. Points on the same side of the plane have the same sign.