In the Clifford algebra, any “unit” vector, i.e. a vector \({u}\) with \({\left\langle u,u\right\rangle =\pm1}\), has inverse \({u^{-1}=u/\left\langle u,u\right\rangle =\pm u}\). It turns out that for any vector \({v}\), the quantity \({R_{u}(v)\equiv-uvu^{-1}}\) is the reflection of \({v}\) in the hyperplane orthogonal to the unit vector \({u}\).

We can see this by decomposing \({v}\) into a part \({v_{\Vert}=\left(\left\langle u,v\right\rangle /\left\langle u,u\right\rangle \right)u}\) that is parallel to \({u}\) and the remaining part \({v_{\perp}}\) that is orthogonal to \({u}\). Since parallel vectors commute, we have \({-uv_{\Vert}u^{-1}=-v_{\Vert}}\). In contrast, orthogonal vectors anti-commute, so that we have \({-uv_{\perp}u^{-1}=v_{\perp}}\), and thus

\(\displaystyle R_{u}\left(v\right)=-uvu^{-1}=-u\left(v_{\perp}+v_{\Vert}\right)u^{-1}=v_{\perp}-v_{\Vert}. \)


An Illustrated Handbook