Definitions

# 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

$\left\{mathbf v\right\}= v_1 boldsymbol\left\{hat\left\{x\right\}\right\} + v_2 boldsymbol\left\{hat\left\{y\right\}\right\} + v_3 boldsymbol\left\{hat\left\{z\right\}\right\}$
where $boldsymbol\left\{hat\left\{x\right\}\right\}, boldsymbol\left\{hat\left\{y\right\}\right\}, boldsymbol\left\{hat\left\{z\right\}\right\}$ is a basis. The the direction cosines are
begin\left\{align\right\}
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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