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Direction cosine

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

{mathbf v}= v_1 boldsymbol{hat{x}} + v_2 boldsymbol{hat{y}} + v_3 boldsymbol{hat{z}}
where boldsymbol{hat{x}}, boldsymbol{hat{y}}, boldsymbol{hat{z}} is a basis. The the direction cosines are
alpha & = cos a = frac{{mathbf v} cdot boldsymbol{hat{x}} }{ left Vert {mathbf v} right Vert }, beta & = cos b = frac{{mathbf v} cdot boldsymbol{hat{y}} }{ left Vert {mathbf v} right Vert }, gamma &= cos c = frac{{mathbf v} cdot boldsymbol{hat{z}} }{ left Vert {mathbf v} right Vert }. end{align}
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