Exercise 4.2.5

$P_1=\frac{1}{9}\left[\begin{array}{ccc}\hfill 1& \hfill -2& \hfill -2\\ \hfill -2& \hfill 4& \hfill 4\\ \hfill -2& \hfill 4& \hfill 4\end{array}\right]$and $P_2=\frac{1}{9}\left[\begin{array}{ccc}\hfill 4& \hfill 4& \hfill -2\\ \hfill 4& \hfill 4& \hfill -2\\ \hfill -2& \hfill -2& \hfill 1\end{array}\right]$.

$P_1$ and $P_2$ are the projection matrices onto the lines through $a_1=(-1,2,2)$ and $a_2=(2,2,-1).P_1{P}_2=$ zero matrix because $a_1\perp a_2$.