Plane through  and perpendicular to the direction (a, b, c):

Example 1

Find the equation for the plane through the point (0, 1, 2) perpendicular to the vector (2, 1, -3).

Solution:

= (0, 1, 2)

(a, b, c) = (2, 1, -3)

The plane:    2 (x - 0) + 1 (y - 1) - 3 (z - 2) = 0

2x + y -3z = -5.

Plane through  ,  and :

Example 2:

Find the equation for the plane through the points (0, 1, 2), (2, 1, 3) and (3, 1, 0)

Solution:

Plane through  and parallel to the vectors  and :

Plane through and and parallel to the direction (a,b,c):

The distance from the point to the plane ax+by+cz+d=0 is

The angle between two planes  and is

Two planes are parallel if their normal vectors are parallel (constant multiples of one another). It is easy to recognize parallel planes written in the form ax+by+cz=d since a quick comparison of the normal vectors n=<a,b,c> can be made.