A Question About Orthogonal Projections

Hey there!

I have my matrix A and b and have found the general solution for $x$ of $Ax=b$. The next question i have is to project the general solution onto the rowspace of $A$. I have the orthogonal projector which i believe is $P=A^T(AA^T)^{-1}A$, im just curious as to how i actually approach the projection. My general solution is $x_g=\begin{bmatrix} 0 \\1 \\0 \end{bmatrix}+t\begin{bmatrix}-0.5 \\-0.5 \\1 \end{bmatrix}$so do i just sub this into $Proj_{rowspace(A)}(x_g) =Px_g$ and work from there? Or is there something else i need to do that im missing?