Direction Cosines and Direction Ratios

A line in space has a definite length only between two specified points, but it has a direction regardless of length -- and describing that direction precisely, independent of where the line is positioned, is the purpose of direction cosines and direction ratios.

Direction cosines. Let a directed line $L$ (or a vector parallel to it) make angles $\alpha$, $\beta$, $\gamma$ with the positive $x$-, $y$- and $z$-axes respectively. The cosines of these three angles, $l=\cos\alpha$, $m=\cos\beta$, $n=\cos\gamma$, are called the direction cosines of the line, written as the ordered triple $(l,m,n)$. Every line has exactly two associated sets of direction cosines, $(l,m,n)$ and $(-l,-m,-n)$, one for each of its two possible directions.

The relation $l^2+m^2+n^2=1$ -- derivation. Consider a directed line through the origin, and let $P(x,y,z)$ be a point on it at distance $r=OP$ from $O$. Since $\alpha$ is the angle $OP$ makes with the $x$-axis, the foot of the perpendicular from $P$ to the $x$-axis lies at distance $x=r\cos\alpha=rl$ from $O$; similarly $y=rm$ and $z=rn$. Substituting into the distance-from-origin formula $r^2=x^2+y^2+z^2$ (Section 3),

(dividing through by $r^2\ne0$). So the direction cosines of any line always satisfy $l^2+m^2+n^2=1$ -- they cannot be chosen freely as three independent numbers, only as a triple lying on the unit sphere.

Direction cosines of the line joining two points. For two points $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$, the direction cosines of the line $P_1P_2$ (directed from $P_1$ to $P_2$) are obtained by dividing each coordinate difference by the distance $P_1P_2$ (Section 3):

This automatically satisfies $l^2+m^2+n^2=1$, since the sum of the squares of the numerators is exactly $P_1P_2^{\,2}$ by the distance formula.

Direction ratios. In practice, a line's direction is very often known only up to scale -- e.g. as coefficients $a,b,c$ in an equation -- without the corresponding distance being known or needed. Any three numbers $a,b,c$ proportional to a line's direction cosines, i.e. $l=\lambda a$, $m=\lambda b$, $n=\lambda c$ for some nonzero constant $\lambda$, are called the line's direction ratios. Direction ratios are far more convenient to work with than direction cosines, since they can be read directly off two points as $x_2-x_1,\ y_2-y_1,\ z_2-z_1$, with no need to compute a square root, and any nonzero scalar multiple of a valid set of direction ratios represents the same direction. …