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12.3 Lines

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).

Distances

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 PP in the ratio k:(100-k). As a particular case, the midpoint of PP 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):

Angles

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 aa+bb+cc=0.

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

arccos( cos cos + cos cos + cos cos )

Concurrence, Coplanarity, Parallelism

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

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.