Math Formulas: Planes in three dimensions

Plane forms

Point direction form:

where P₁(x₁,y₁,z₁) lies in the plane, and the direction (a,b,c) is normal to the plane.

General form:

where direction (A,B,C) is normal to the plane.

Intercept form:

this plane passes through the points (a,0,0), (0,b,0), and (0,0,c).

Three point form:

Normal form:

Parametric form:

where the directions (a₁,b₁,c₁) and (a₂,b₂,c₂) are parallel to the plane.

Angle between two planes:

is

The planes are parallel if and only if

Equation of a plane

The equation of a plane through P₁(x₁,y₁,z₁) and parallel to directions (a₁,b₁,c₁) and (a₂,b₂,c₂) has equation

The equation of a plane through P₁(x₁,y₁,z₁) and P₂(x₂,y₂,z₂), and parallel to direction (a,b,c), has equation

Distance

The distance of P₁(x₁,y₁,z₁) from the plane Ax + By + Cz + D = 0 is

Intersection

The intersection of two planes

is the line

where

If a = b = c = 0, then the planes are parallel.