Suppose that \(\vu\) and \(\vv\) are vectors in \(\R^n\text{.}\) We wish to find \(\vw_1\) and \(\vw_2\) so that \(\vw_1\) is parallel to \(\vv\) and \(\vw_2\) is orthogonal to \(\vv\text{,}\) as shown in Figure 9.3.8.
Compute \(\vu\cdot \vv\) as \((\vw_1+\vw_2)\cdot \vv\text{.}\) Simplify your answer as much as possible, using the fact that \(\vw_2\) is orthogonal to \(\vv\text{.}\)
Since \(\vw_1\) is parallel to \(\vv\text{,}\) there is a scalar \(k\) so that \(\vw_1 = k\vv\text{.}\) Substitute \(k\vv\) for \(\vw_1\) in your answer to the previous part and then solve for \(k\text{.}\)
