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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

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.

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.

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**.)

*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.