Let $H=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \mathbf{u} \cdot \mathbf{x}=c\right\}$ be a hyperplane in $\mathbb{R}^{n}$, where $\mathbf{u}$ is a unit vector and $c$ is a constant. Show that the reflection map

$$\mathbf{x} \mapsto \mathbf{x}-2(\mathbf{u} \cdot \mathbf{x}-c) \mathbf{u}$$

is an isometry of $\mathbb{R}^{n}$ which fixes $H$ pointwise.

Let $\mathbf{p}, \mathbf{q}$ be distinct points in $\mathbb{R}^{n}$. Show that there is a unique reflection $R$ mapping $\mathbf{p}$ to $\mathbf{q}$, and that $R \in O(n)$ if and only if $\mathbf{p}$ and $\mathbf{q}$ are equidistant from the origin.

Show that every isometry of $\mathbb{R}^{n}$ can be written as a product of at most $n+1$ reflections. Give an example of an isometry of $\mathbb{R}^{2}$ which cannot be written as a product of fewer than 3 reflections.