Orthogonality & Least Squares · Medium problem

Sensor fusion combines three direction readings into a $5\times3$ matrix $A$ whose columns are $(1,1,0,0,0)$, $(0,1,1,0,0)$, and $(1,2,1,0,0)$, and builds the orthogonal projection matrix $P$ onto the column space of $A$. Find $\operatorname{tr}(P)$, the effective number of independent directions actually captured.