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Section 11.3 : Dot Product

10. Given \(\vec u = 7\vec i - \vec j + \vec k\) and \(\vec w = - 2\vec i + 5\vec j - 6\vec k\) compute \({{\mathop{\rm proj}\nolimits} _{\,\vec w}}\,\vec u\).

Show Solution

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

\[\vec u\centerdot \vec w = - 25\hspace{0.25in}\hspace{0.25in}{\left\| {\vec w} \right\|^2} = 65\]

The projection is then,

\[{{\mathop{\rm proj}\nolimits} _{\,\vec w}}\,\vec u = \frac{{ - 25}}{{65}}\left( { - 2\vec i + 5\vec j - 6\vec k} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{10}}{{13}}\vec i - \frac{{25}}{{13}}\vec j + \frac{{30}}{{13}}\vec k}}\]