PlanetMath: direction cosines

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direction cosines

(Definition)

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 -axis, -axis and -axis, respectively, then the numbers

$\displaystyle \cos{\alpha},\; \cos{\beta},\; \cos{\gamma}$

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

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

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

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

We also see that the direction cosines satisfy

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

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"direction cosines" is owned by pahio. [ full author list (2) ]

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Also defines:

direction numbers

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Attachments:: equation of plane (Topic) by pahio

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Cross-references: coordinate, components, vector, numbers, positive, angles, non-zero vector
There are 4 references to this entry.

This is version 4 of direction cosines, born on 2007-06-18, modified 2007-08-17.
Object id is 9615, canonical name is DirectionCosines.
Accessed 1472 times total.

Classification:

AMS MSC:	15A72 (Linear and multilinear algebra; matrix theory :: Vector and tensor algebra, theory of invariants)
	51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

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