If the non-zero vector $\vec{r} = x\vec{i}+y\vec{j}+z\vec{k}$ , of $\mathbb{R}^3$ 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 to the direction cosines, are direction numbers of the vector.

If $r = \sqrt{x^2+y^2+z^2}$ , is the length of $\vec{r}$ 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 $$\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1.$$