# 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 d 0 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 d 0, the same as the sign of c when d=0 and c 0, 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 a x+b y+c z+d =0 and a x+b y+c z+d =0 is In particular, the two planes are parallel when a :b :c = a :b :c , and perpendicular when a a +b b +c c =0.

### Concurrence and Coplanarity

Four planes a x+b y+c z+d =0, a x+b y+c z+d =0, a x+b y+c z+d =0 and a x+b y+c z+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.)

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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.