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### Section 11.3 : Dot Product

11. Determine the direction cosines and direction angles for $$\displaystyle \vec r = \left\langle { - 3, - \frac{1}{4},1} \right\rangle$$.

Show Solution

All we really need to do here is use the formulas from the notes. That will need the following quantity.

$\left\| {\vec r} \right\| = \sqrt {\frac{{161}}{{16}}} = \frac{{\sqrt {161} }}{4}$

The direction cosines and angles are then,

$\cos \alpha = \frac{{ - 3}}{{{}^{{\sqrt {161} }}/{}_{4}}} = - \frac{{12}}{{\sqrt {161} }}\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}\alpha = {\cos ^{ - 1}}\left( { - \frac{{12}}{{\sqrt {161} }}} \right) = 2.8106\,\,{\rm{radians}}$ $\cos \beta = \frac{{ - {}^{1}/{}_{4}}}{{{}^{{\sqrt {161} }}/{}_{4}}} = - \frac{1}{{\sqrt {161} }}\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}\beta = {\cos ^{ - 1}}\left( { - \frac{1}{{\sqrt {161} }}} \right) = 1.6497\,\,{\rm{radians}}$ $\cos \gamma = \frac{1}{{{}^{{\sqrt {161} }}/{}_{4}}} = \frac{4}{{\sqrt {161} }}\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}\gamma = {\cos ^{ - 1}}\left( {\frac{4}{{\sqrt {161} }}} \right) = 1.2501\,\,{\rm{radians}}$