rotation of a line to the $y$ axis

Question

for the line, $4x -3y = 0$, find the matrix for a counterclockwise rotation taking the line $4x-3y=0$ to the $y$-axis.

I have no idea what this question is even asking. Is it asking for a matrix that would rotate the line parallel to the $y$ axis?

If so, how do I go about doing that? I can see that it is a pythagorean triple line, and I can sort of see how those numbers $(3,4,5)$ could be used to get the answer, but after that I'm lost.

Answer 1

There are so many ways to find the matrix.

Probably the easiest way is to observe that all rotation matrix are of the form

$$\begin{bmatrix} \cos \theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix},$$

where $\theta$ is angle that you want to turn (counter-clockwisely). Thus you need only to find the angle between that line and the $y$-axis.