Reflections in Euclidean plane Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the counterclockwise rotation of $\frac{\pi}{2}$ and $S: \mathbb{R}^2 \to \mathbb{R}^2$ be the reflection w.r.t. the line $x+3y=0$. There exists a reflection $R$ such that $T^{-1}ST=R$? Is there a canonical way to find which is the line w.r.t. we are reflection through $R$?

suppose $T^{-1}ST $ fixes a line $l.$ then $$T^{-1}ST(l) = l \implies ST(l) = T(l). $$  that is $T(l)$ is fixed by $S$ and we know that only lines fixed by $S$ is the line $3x+y = 0.$  therefore $T(l)$ is $y-3x = 0$ and $$  T^{-1}ST \text{ is a reflection on the line } y - 3x = 0.$$
