# 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

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

In the preceding figure, define $${\hat{v}_{\perp}}$$ to be the unit vector perpendicular to $${u}$$; then

\begin{aligned}u^{\prime} & \equiv u\cos\frac{\theta}{2}+\hat{v}_{\perp}\sin\frac{\theta}{2}\end{aligned}

is a vector rotated by $${\theta/2}$$ from $${u}$$. The combination of reflections using these two vectors yields a rotation of $${v}$$ by $${\theta}$$ in the $${u\wedge v}$$ plane:

\begin{aligned}R_{\theta}\left(v\right) & =u^{\prime}uvuu^{\prime}\\
& =\left(u^{\prime}u\right)v\left(u^{\prime}u\right)^{-1}
\end{aligned}

Note that for infinitesimal $${\theta}$$, we then have

\begin{aligned}u^{\prime}u & =1+\hat{v}_{\perp}u\theta/2\\
& =\exp\left(\hat{v}_{\perp}u\theta/2\right),
\end{aligned}

so that an infinitesimal rotation corresponds to the exponential of a bivector.