Let $x_1, x_2, x_3$ be coordinates of a positively orientated orthonormal basis $f_1, f_2, f_3$. Find the matrix that corresponds to an orthogonal projection onto the plane $$x_1 + x_2 + 3x_3 = 0$$ and that is also parallel to the line $u(1,1,-1)$, where $u \in \mathbb{R}$.
I am aware that the plane has a normal $n$ such that $n=\frac{1}{\sqrt{11}}(1,1,3)$. Let $x=(x_1, x_2,x_3)$, then the projection, $p$, onto the plane should be: $p=x\ - \langle x, n \rangle\ n$, which gives me the matrix
$$\frac{1}{11}\begin{bmatrix}\ 10&-1&-3\\-1&10&-3\\-3&-3&2 \end{bmatrix}$$ I don't think my solution is correct, since I don't even know how to make sure that this somehow yields a vector that is parallell to $(1,1,-1)$. How is one supposed to solve excercises like this one?