Direction Cosines and Direction Ratios

Direction angles. Let $P(x,y,z)$ be a point in space at distance $r=\sqrt{x^2+y^2+z^2}$ from the origin $O$, and let $R,S,T$ be the feet of the perpendiculars from $P$ to the $x,y,z$ axes. Let $\alpha,\beta,\gamma$ be the angles that $\vec{OP}$ makes with the positive $x,y,z$ axes respectively — these are the direction angles of $\vec{OP}$.

Direction cosines. In right triangle $OPR$ (right-angled at $R$), $\cos\alpha=\dfrac{OR}{OP}=\dfrac xr$; similarly $\cos\beta=\dfrac yr,\ \cos\gamma=\dfrac zr$. These three cosines, usually denoted $(l,m,n)$, are the direction cosines of $\vec{OP}=x\hat i+y\hat j+z\hat k$: $l=\cos\alpha=\frac xr,\quad m=\cos\beta=\frac yr,\quad n=\cos\gamma=\frac zr,\qquad r=\sqrt{x^2+y^2+z^2}.$

Direction ratios. Any three numbers proportional to the direction cosines of a vector are called its direction ratios. Since 'proportional to' allows any nonzero scaling, a vector has infinitely many possible sets of direction ratios — unlike its direction cosines, which are essentially fixed (up to an overall sign, corresponding to the two opposite directions along the same line).

A vector not through the origin. If a vector's initial point is not the origin, we find its direction cosines by translating a copy of it (same magnitude, same direction) so that its tail sits at the origin — two equal vectors necessarily share the same set of direction cosines.

Key results. Let $\vec r=x\hat i+y\hat j+z\hat k$ have direction angles $\alpha,\beta,\gamma$. Then:

1. $l^2+m^2+n^2=1$ (the sum of the squares of the direction cosines is always $1$). Proof. $l^2+m^2+n^2=\dfrac{x^2}{r^2}+\dfrac{y^2}{r^2}+\dfrac{z^2}{r^2}=\dfrac{x^2+y^2+z^2}{r^2}=\dfrac{r^2}{r^2}=1$.
2. $\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2$ (immediate from (i), since $\sin^2\theta=1-\cos^2\theta$ for each angle, and the three $1$'s sum to $3$, minus the $1$ from (i), leaves $2$).
3. The direction cosines of $\vec r$ are $\left(\frac{x}{\sqrt{x^2+y^2+z^2}},\ \frac{y}{\sqrt{x^2+y^2+z^2}},\ \frac{z}{\sqrt{x^2+y^2+z^2}}\right).$
4. $l,m,n$ are the direction cosines of some vector if and only if $l^2+m^2+n^2=1$ — this is exactly the test used to check whether a given triple of numbers could be direction cosines. …