reflection of a sphere

Given the n-sphere, the \(k\)-th reflection for \(k \in \{1,...,n+1\}\) is defined as continuous map

\begin{align*} r_{k}: S^n \rightarrow& S^n \\ (s_1,...,s_k,...,s_{n+1}) \mapsto& (s_1,...,-s_k,...,s_{n+1}) \end{align*}