# I need to find the matrix of reflection through line y=-(2)/(3)x

I need to find the matrix of reflection through line $y=-\frac{2}{3}x$
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?
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eri1ti0m
Alternatively to the comment above (which requires a bit of trig), where does your matrix T send $\left(\begin{array}{c}3\\ -2\end{array}\right)$ and $\left(\begin{array}{c}2\\ 3\end{array}\right)$ which are on and perpendicular to your line respectively. Now can you find a and b in terms of x and y so that
$\left(\begin{array}{c}x\\ y\end{array}\right)=a\left(\begin{array}{c}3\\ -2\end{array}\right)+b\left(\begin{array}{c}2\\ 3\end{array}\right)$
Finally, apply T to find T$\left(\begin{array}{c}x\\ y\end{array}\right)$
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Avery Stewart
Notice that vectors on this line have the form $\left(1,-\frac{2}{3}\right)$, and an orthogonal vector would be $\left(\frac{2}{3},1\right)$. A very straightforward procedure could be to reflect the vectors $\left(0,1\right)$ and $\left(1,0\right)$ orthogonally in this line. Once you have determined the images of the basis vectors, you can figure out what the matrix columns should look like.