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Let $\tau_{w}$ be any translation and let $\ell = P + [w^{\perp}]$ be any line having $w$ as normal vector. Show that if $m = P - \dfrac{1}{2} w + [w^{\perp}]$ and $m^{\prime} = P + \dfrac{1}{2} w + [w^{\perp}]$, then

$$\Omega_{\ell} \Omega_{m} = \Omega_{m^{\prime}} \Omega_{\ell} = \tau_{w}$$

I need some light on this. Thanks. :]

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    By "let $\tau_w$ be any translation", do you really mean "let $w$ be any vector and let $\tau_w$ be the translation along $w$"? What is our space? Is it, for instance, finite-dimensional? And what is $\Omega$? – 2017-01-04
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    Basically, you’re being asked to show that any translation can be expressed as the composition of a pair of reflections. They’ve even given you the lines about which to reflect, so all you have to do is work out those reflections and compute their products. – 2017-01-04

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