$W$ is the subspace spanned by the vector $(3,4)$. Hence, any vector in $W$ has the form $w=(3c,4c)$, where $c$ is a scalar. I want to find a formula for the orthogonal projection of $\mathbb{R}^2$ onto this subspace. Any vector in $\mathbb{R}^2$ can be written in the form, $v=(a,b)$. We have that the orthogonal projection of $v$ onto $w$ is given by, $$\frac{w\cdot v}{w\cdot w}w.$$ So we have $$proj_w(v)=\frac{(3a+4b)c^2}{(25c^2)}(3,4)=\frac{3a+4b}{25}(3,4).$$ I also want to find the matrix of this projection in the standard basis for $\mathbb{R}^2$. We do this by seeing how the standard basis transforms under this projection. We have $$proj_w(\epsilon_1)=\frac{3}{25}(3,4),$$ and $$proj_w(\epsilon_2)=\frac{4}{25}(3,4).$$ So in the standard basis the matrix is $$\frac{1}{25}\begin{bmatrix}9 & 12 \\ 12 & 16\end{bmatrix}.$$ Am I on the right track?
I'm also seeking $W^{\perp}$, which I'm told is the null space. Row-reducing our matrix we get $$\begin{bmatrix}3 & 4 \\ 0 & 0\end{bmatrix}.$$ So the null space is the set of vectors such that $3a=-4b$?