Let \(\vv = \langle 4, -8 \rangle\text{.}\) Find \(\proj_{\vv} \vu\text{,}\)\(\comp_{\vv} \vu\text{,}\) and \(\proj_{\perp \vv} \vu\text{.}\) Draw a picture to illustrate the vectors involved. Finally, express \(\vu\) as the sum of two vectors where one is parallel to \(\vv\) and the other is perpendicular to \(\vv\text{.}\)
Now let \(\vw = \langle -2,4 \rangle \text{.}\) Add \(\vw\) to the picture you drew in the previous part. Without doing any calculations, find \(\proj_{\vw} \vu\text{.}\) Explain your reasoning.
Find a vector \(\vw\) not parallel to \(\vz = \langle 3,4 \rangle \) such that \(\proj_{\vz} \vw\) has length \(10\text{.}\) Note that there are infinitely many different answers!
