Orthogonal Projection of plane onto a line

Question

Show that the orthogonal projection of the plane onto the line that makes an angle θ with the x axis is given by the matrix:

$\begin{bmatrix}\cos^2 \theta & \sin\theta\cos\theta \\ \sin\theta\cos\theta & \sin^2 \theta \end{bmatrix}$

I've looked around for a while and I can't find any solution or answer that points me in the right direction. How do I do this question?

Answer 1

Find the point along the line $(t\cos\theta,t\sin\theta)$ that forms a right angle with the origin and the point to be projected.

$$(t\cos\theta,t\sin\theta)\cdot((x,y)-(t\cos\theta,t\sin\theta))=0.$$

We draw

$$t(x\cos\theta+y\sin\theta)-t^2=0$$

or

$$t=x\cos\theta+y\sin\theta.$$

Now the projection is

$$((x\cos\theta+y\sin\theta)\cos\theta,(x\cos\theta+y\sin\theta)\sin\theta).$$