direction cosines

If the non-zero vector  $\overrightarrow{r}=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$  of ${ℝ}^3$ forms the angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of $x$-axis, $y$-axis and $z$-axis, respectively, then the numbers

$$\cos \alpha ,\cos \beta ,\cos \gamma$$

are the direction cosines of the vector. Any triple $l,m,n$ of numbers, which are proportional (http://planetmath.org/Variation) to the direction cosines, are direction numbers of the vector.

If  $r=\sqrt{x^2+y^2+z^2}$  is the of $\overrightarrow{r}$, we see easily that

$$\cos \alpha =\frac{x}{r},\cos \beta =\frac{y}{r},\cos \gamma =\frac{z}{r}.$$

Conversely, the components of the vector on the coordinate axes may be obtained from

$$x=r\cos \alpha ,y=r\cos \beta ,z=r\cos \gamma .$$

We also see that the direction cosines satisfy

$${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1.$$