Direction Cosines and Direction Ratios

Definition of Direction Cosines

Let a vector $\vec a$ (or the directed line $OP$ from the origin $O$ to a point $P$) make angles $\alpha,\beta,\gamma$ with the positive $x$-, $y$- and $z$-axes respectively. The cosines of these angles, $\cos\alpha,\cos\beta,\cos\gamma$, are called the direction cosines of the vector, usually denoted $l,m,n$:

Direction cosines describe the direction of a vector only, independent of its length -- a vector and any positive scalar multiple of it (pointing the same way) share the same direction cosines.

The Fundamental Relation $l^2+m^2+n^2=1$

Derivation. Consider a vector $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$ of magnitude $|\vec a|=r=\sqrt{a_1^2+a_2^2+a_3^2}$, drawn from the origin to the point $P(a_1,a_2,a_3)$. Drop a perpendicular from $P$ to the $x$-axis, meeting it at $N$; then $ON=a_1$, and in the right triangle $OPN$, $\angle PON=\alpha$, so

By the same argument using the perpendiculars to the $y$- and $z$-axes,

Hence $l=\dfrac{a_1}{r}$, $m=\dfrac{a_2}{r}$, $n=\dfrac{a_3}{r}$. Squaring and adding all three,

Direction Ratios

Any three numbers $a,b,c$ proportional to the direction cosines $l,m,n$ of a vector (i.e. $\dfrac{a}{l}=\dfrac{b}{m}=\dfrac{c}{n}$ for some nonzero constant) are called the direction ratios of that vector. Unlike direction cosines, direction ratios are not unique or normalised -- $(2,-1,2)$ and $(4,-2,4)$ describe the same direction -- and need not satisfy $a^2+b^2+c^2=1$.

Given direction ratios $a,b,c$, the direction cosines are recovered by dividing each by $\sqrt{a^2+b^2+c^2}$:

(taking the negative square root instead gives the direction cosines of the oppositely-directed vector.)

Direction Cosines of the Vector Joining Two Points …