** Next:** 13 Polyhedra
**Up:** 12 DirectionsPlanes and Lines
** Previous:** 12.2 Planes

Two planes that are not parallel or coincident intersect in a **straight
line**, so one can express a line by a pair of linear equations

*ax*+*by*+*cz*+*d*=0 and *a'x*+*b'y*+*c'z*+*d'*=0

such that *bc'*-*cb'*, *ca'*-*ac'*, and *ab'*-*ba'* are not all zero.
The line thus defined is parallel to the vector

(*bc'*-*cb'*, *ca'*-*ac'*, *ab'*-*ba'*).

The **direction cosines** of the line are those of this vector: see
(12.1.1)
. (The direction cosines of a line are only
defined up to a simultaneous change in sign, since the opposite vector
still gives the same line.)

The following particular cases are important:

Line through (*x*,*y*,*z*) parallel to the vector (*a*,*b*,*c*):

Line through (*x*,*y*,*z*) and (*x*,*y*,*z*):

This line is parallel to the vector (*x*-*x*, *y*-*y*,
*z*-*z*).

The **distance** between two points in space is the **length
of the line segment** joining them. The distance between the points
(*x*,*y*,*z*) and (*x*,*y*,*z*) is
is

The point *k*% of the way from *P*=(*x*,*y*,*z*)
to *P*=(*x*,*y*,*z*) is

(The same formula works also in oblique coordinates.) This point
divides the segment *P**P* in the ratio *k*:(100-*k*). As a
particular case, the **midpoint** of *P**P* is given by

(½(*x*+*x*),
½(*y*+*y*),
½(*z*+*z*)).

The **Distance** between the point (*x*,*y*,*z*) and the line
through (*x*,*y*,*z*) in direction (*a*,*b*,*c*):

**Distance** between
the line through (*x*,*y*,*z*) in direction (*a*,*b*,*c*) and
the line through (*x*,*y*,*z*) in direction (*a*,*b*,*c*):

**Angle** between lines with directions (*x*,*y*,*z*) and
(*x*,*y*,*z*):

In particular, the two lines are **parallel** when *a*:*b*:*c*=
*a*:*b*:*c*, and **perpendicular** when *a**a*+*b**b*+*c**c*=0.

**Angle** between lines with direction angles
,, and
,,:

arccos( cos cos + cos cos + cos cos )

Two lines specified by point and direction are **coplanar** if and
only if the determinant in the numerator of (1)
is
zero. In this case they are **concurrent** (if the denominator is
nonzero) or **parallel** (if the denominator is zero).

Three lines with directions (*a*,*b*,*c*), (*a*,*b*,*c*) and
(*a*,*b*,*c*) are **parallel to a common plane** if and only if

*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.