Reflection through a line

Question

I need to find the matrix of reflection through line $y=- \frac 23 x$ .

I'm trying to visualise a vector satisfying this. The standard algorithm states that we need to find the angle this line makes with $x$ axis and the transformation matrix can be seen as $R_\alpha T_0 R_{-\alpha}.$

I'm not sure how to proceed. I can't visualise the angle it makes with $x$ axis. Is there a procedure to think about such reflections?

Answer 1

Notice that vectors on this line have the form $(1, -\frac{2}{3} )$, and an orthogonal vector would be $(\frac{2}{3}, 1) $. A very straightforward procedure could be to reflect the vectors $(0,1)$ and $(1,0)$ orthogonally in this line. Have you done this before? Once you have determined the images of the basis vectors, you can figure out what the matrix columns should look like.

Answer 2

Alternatively to the comment above (which requires a bit of trig), where does your matrix T send $\left( \begin{matrix} 3 \\ -2 \\ \end{matrix} \right)$ and $\left( \begin{matrix} 2 \\ 3 \\ \end{matrix} \right)$ which are on and perpendicular to your line respectively. Now can you find a and b in terms of x and y so that $$\left( \begin{matrix} x \\ y \\ \end{matrix} \right)=a\left( \begin{matrix} 3 \\ -2 \\ \end{matrix} \right)+b\left( \begin{matrix} 2 \\ 3 \\ \end{matrix} \right)$$ Finally, apply T to find T$\left( \begin{matrix} x \\ y \\ \end{matrix} \right)$