# direction cosines

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 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 $\vec{r}$, 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

 $\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=1.$
Title direction cosines DirectionCosines 2013-03-22 17:16:32 2013-03-22 17:16:32 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 15A72 msc 51N20 MutualPositionsOfVectors EquationOfPlane direction numbers