direction cosinesDirectionCosines2007-06-18 09:42:332008-11-27 12:43:40Definitionposition vectordirection numbers% this is the default PlanetMath preamble. as your knowledge
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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 {\em direction cosines} of the vector. Any triple $l,\,m,\,n$ of numbers, which are \PMlinkname{proportional}{Variation} to the direction cosines, are {\em direction numbers} of the vector.
If\, $r = \sqrt{x^2+y^2+z^2}$\, is the \PMlinkescapetext{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
$$\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1.$$