Direction cosines and direction ratios

Direction cosines and direction ratios of a line

The direction cosines of a line: The direction cosines of a line are defined as the direction cosines of any vector whose support is the given line.

If A and B are two points on a given line L, then direction cosines of vectors AB and BA are the direction cosines of line L. If α, β, γ, are the angles which the line L makes with positive directions of x-axis, y-axis, and z-axis respectively, then its direction cosines are either cos α, cos β , cos γ or -cos α, -cos β ,- cos γ.

Therefore if l,m,n are direction cosines of a line, then –l.-m,-n are also its direction cosines.

Also l² +m² + n² = 1

If P and Q are two points with coordinates P(x1,y1,z1) and Q(x2,y2,z2) on line L, it direction cosines (of L or PQ) are

(x2-x1)/PQ, (y2-y1)/PQ, , (z2-z1)/PQ, or (x1-x2)/PQ, (y1-y2)/PQ, , (z1-z2)/PQ