Given a vector $\mathbf{x}=\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}$, write down the vector $\mathbf{x}^{\prime}$ obtained by rotating $\mathbf{x}$ through an angle $\theta$.

Given a unit vector $\mathbf{n} \in \mathbb{R}^{3}$, any vector $\mathbf{x} \in \mathbb{R}^{3}$ may be written as $\mathbf{x}=\mathbf{x}_{\|}+\mathbf{x}_{\perp}$ where $\mathbf{x}_{\|}$is parallel to $\mathbf{n}$ and $\mathbf{x}_{\perp}$ is perpendicular to $\mathbf{n}$. Write down explicit formulae for $\mathbf{x}_{\|}$and $\mathbf{x}_{\perp}$, in terms of $\mathbf{n}$ and $\mathbf{x}$. Hence, or otherwise, show that the linear map

$$\mathbf{x} \mapsto \mathbf{x}^{\prime}=(\mathbf{x} \cdot \mathbf{n}) \mathbf{n}+\cos \theta(\mathbf{x}-(\mathbf{x} \cdot \mathbf{n}) \mathbf{n})+\sin \theta(\mathbf{n} \times \mathbf{x})$$

describes a rotation about $\mathbf{n}$ through an angle $\theta$, in the positive sense defined by the right hand rule.

Write equation $(*)$ in matrix form, $x_{i}^{\prime}=R_{i j} x_{j}$. Show that the trace $R_{i i}=1+2 \cos \theta$.

Given the rotation matrix

$$R=\frac{1}{2}\left(\begin{array}{ccc} 1+r & 1-r & 1 \\ 1-r & 1+r & -1 \\ -1 & 1 & 2 r \end{array}\right)$$

where $r=1 / \sqrt{2}$, find the two pairs $(\theta, \mathbf{n})$, with $-\pi \leqslant \theta<\pi$, giving rise to $R$. Explain why both represent the same rotation.