# prove that distinct reflections commute iff...

let $m$ and $l$ be lines in $E_2$, and the reflection of $l$ is the mapping $\Omega_l$ of $E_2$ to $E_2$ defined by $\Omega_lX = X - 2N((X - P) \cdot N)$ where $N$ is the unit normal to $l$ and $P$ is any point on $l$
Show that two distinct reflections $\Omega_l$ and $\Omega_m$commute if and only if $m \perp l$.