Consider two unit vectors $u, v \in V$. Prove that $R_{\frac{u-v}{|u-v|}}$ sends $u$ to $v$ and $v$ to $u$.
Here $R_{\frac{u-v}{|u-v|}}$ is a reflection, in my book it is defined as $$ R_{m} m=-m \text { and } R_{m} v=v \text { for all } v \perp m $$ Or it can be written explicitly as $$ R_{m} v=v-2 m\langle m, v\rangle $$ I tried using the explicit formula but I did not obtain any useful results.
Define $w=\frac{u-v}{|u-v|}$. Then according to the formula for $R$,
$$ R_wu=u-2w\langle w,u\rangle $$
$$ = u-2\frac{u-v}{|u-v|}\left\langle\frac{u-v}{|u-v|},u\right\rangle $$
$$ = u-2(u-v)\frac{\langle u-v,u\rangle}{|u-v|^2} $$
$$ = u - 2(u-v)\frac{1-\langle u,v\rangle}{|u-v|^2}. $$
Now, $|u-v|^2=|u|^2-2\langle u,v\rangle+|v|^2=2(1-\langle u,v\rangle)$, so after cancelling,
$$ = u-2(u-v)(\frac{1}{2}) = v. $$
I'll let you show $R_wv=u$ similarly.