Paul's Online Notes
Home / Calculus II / Vectors / Dot Product
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 5-3 : 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,

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