Suppose \(\mathbf{R}\in \mathrm{SO}(3)\) is a rotation around some axis with rotation angle \(\vartheta.\) Show that \(\cos \vartheta=\frac{1}{2}(\operatorname{Tr}(\mathbf{R})-1).\)

Show that all \(\mathbf{R}\in \mathrm{SO}(3)\) have \(1\) as an eigenvalue.

Hint: Use that every cubic polynomial \(p \in \mathsf P_3(\mathbb{R})\) admits a zero.

Show that every orthogonal transformation \(f:\mathbb E^n\to\mathbb E^n\) can be written as a composition of at most \(n\) reflections along hyperplanes.

A map \(f:\mathbb E^n\to\mathbb E^n\) is called an isometry if \(\|\vec v-\vec w\| = \|f(\vec v)-f(\vec w)\|\) for all \(\vec v,\vec w\in \mathbb E^n.\) Show that every isometry that fixes the origin, i.e. \(f(0_{\mathbb E^n}) =0_{\mathbb E^n},\) is an orthogonal transformation.