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12.2 Planes

The (cartesian) equation of a plane is linear in the coordinates x and y, that is, of the form ax+by+cz+d=0. The normal direction to this plane is (a,b,c). The intersection of this plane with the x-axis, or x-intercept, is x=-d/a; the y-intercept is y=-d/b, and the z-intercept is z=-d/c. The plane is vertical (perpendicular to the xy-plane) if c=0; it is perpendicular to the x-axis if b=c=0; and likewise for the other coordinates.

When a+b+c=1 and d0 in the equation ax+by+cz+d=0, the equation is said to be in normal form. In this case d is the distance of the plane to the origin, and (a,b,c) are the direction cosines of the normal.

To reduce an arbitrary equation ax+by+cz+d=0 to normal form, divide by , where the sign of the radical is chosen opposite the sign of d when d0, the same as the sign of c when d=0 and c0, and the same as the sign of b otherwise.

Planes with prescribed properties

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

a(x-x)+b(y-y)+c(z-z)=0

Plane through (x,y,z) and parallel to the directions (a,b,c) and (a,b,c):

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

Plane going through (x,y,z), (x,y,z) and (x,y,z):

(The last three formulas remain true in oblique coordinates.)

The distance from the point (x,y,z) to the plane ax+by+cz+d=0 is

The angle between two planes ax+by+cz+d=0 and ax+by+cz+d=0 is

In particular, the two planes are parallel when a:b:c= a:b:c, and perpendicular when aa+bb+cc=0.

Concurrence and Coplanarity

Four planes ax+by+cz+d=0, ax+by+cz+d=0, ax+by+cz+d=0 and ax+by+cz+d=0 are concurrent if and only if

Four points (x,y,z), (x,y,z), (x,y,z) and (x,y,z) are coplanar if and only if

(Both of these assertions remain true in oblique coordinates.)

The Geometry Center Home Page

Silvio Levy
Wed Oct 4 16:41:25 PDT 1995

This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.