# Reflections

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

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