Answer
See solution
Work Step by Step
The columns of U are $\vec{u_1}$ and $\vec{u_2}$
$U^TU=\begin{bmatrix}
1&0\\0&1
\end{bmatrix}$
$UU^T=\begin{bmatrix}
8/9&-2/9&2/9\\-2/9&5/9&4/9\\2/9&4/9&5/9\\
\end{bmatrix}$
$proj_Wy=6\vec{u_1}+3\vec{u_2}=\begin{bmatrix}
2\\4\\5
\end{bmatrix}$
$UU^Ty=\begin{bmatrix}
8/9&-2/9&2/9\\-2/9&5/9&4/9\\2/9&4/9&5/9\\
\end{bmatrix}\begin{bmatrix}
4\\8\\1
\end{bmatrix}=\begin{bmatrix}
2\\4\\5
\end{bmatrix}$
