Please note brackets () denote inner product - for some reason the traditional symbols for inner product do not show up when I type them in!
Hi,
I am trying to prove that the two reflections in $\mathbf{R}^2$, defined:
$$\begin{align} \Omega_\ell &= X-2(X-P,N)N \\ \Omega_m &= X-2(X-Q,N^\prime)N^\prime \end{align}$$
Commute if and only if the lines $\ell$ and $m$ are perpendicular.
$$\Omega_\ell \circ \Omega_m = \Omega_m \circ \Omega_\ell$$
Using the two algebraic definitions above and simplifying, and then setting the two results equal, I find this implies:
$$\begin{align} 4(X-P,N)(N,N^\prime)N^\prime &= 4(X-Q,N^\prime)(N,N^\prime)N \\[4pt] \quad\implies\quad(X-P,N)(N,N^\prime)N^\prime &=\phantom{4}(X-Q,N^\prime)(N,N^\prime)N \end{align}$$
I've tried everything I can think of to prove that this is true if and only if $\ell$ and $m$ are perpendicular (and thus $N$ and $N^\prime$ are perpendicular). I've tried proof by contraction (assume $N$ and $N^\prime$ aren't perpendicular, ie. $(N,N^\prime) \neq 0$) but haven't gotten anywhere. It is trivial to prove that they are equal (ie. both $0$) if $N$ and $N^\prime$ are perpendicular, but this does not show that this must be the case.
Any help on this problem would be appreciated.
Thank you in advance.