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Section 5-3 : Dot Product

9. Given \(\vec a = \left\langle { - 8,2} \right\rangle \) and \(\vec b = \left\langle { - 1, - 7} \right\rangle \) compute \({{\mathop{\rm proj}\nolimits} _{\,\vec a}}\,\vec b\).

Show Solution

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

\[\vec a\centerdot \vec b = - 6\hspace{0.25in}\hspace{0.25in}{\left\| {\vec a} \right\|^2} = 68\]

The projection is then,

\[{{\mathop{\rm proj}\nolimits} _{\,\vec a}}\,\vec b = \frac{{ - 6}}{{68}}\left\langle { - 8,2} \right\rangle = \require{bbox} \bbox[2pt,border:1px solid black]{{\left\langle {\frac{{12}}{{17}}, - \frac{3}{{17}}} \right\rangle }}\]