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Example:Finding Direction Cosines for given Direction Ratios

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Direction cosine


In analytic geometry, the direction cosines of a vector are the cosines of the angles between the vector and the three coordinate axes.

If v is a vector


where  is a basis. Then the direction cosines are

(Our solved example in mathguru.com uses this concept)

Note that

α2 + β2 + γ2 = 1


(αβγ) is the Cartesian coordinates of the unit vector 


More generally, direction cosine refers to the cosine of the angle between any two vectors. They are useful for forming direction cosine matrices that express one set of orthonormalbasis vectors in terms of another set, or for expressing a known vector in a different basis.




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.


(Our solved example in mathguru.com uses the below concept. This is our own explanation, it is not taken from Wikipedia.)


Direction ratios


For any vector r = a + b+ c its direction ratios are a : b : c. Its direction cosines are

l = a / (a2 +b2 +c2)

m = a / (a2 +b2 +c2)

n = a / (a2 +b2 +c2)