Let $\mathcal{M}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be the linear map defined by

$$\mathbf{x} \mapsto \mathbf{x}^{\prime}=a \mathbf{x}+b(\mathbf{n} \times \mathbf{x})$$

where $a$ and $b$ are positive scalar constants, and $\mathbf{n}$ is a unit vector.

(i) By considering the effect of $\mathcal{M}$ on $\mathbf{n}$ and on a vector orthogonal to $\mathbf{n}$, describe geometrically the action of $\mathcal{M}$.

(ii) Express the map $\mathcal{M}$ as a matrix $M$ using suffix notation. Find $a, b$ and $\mathbf{n}$ in the case

$$M=\left(\begin{array}{rrr} 2 & -2 & 2 \\ 2 & 2 & -1 \\ -2 & 1 & 2 \end{array}\right)$$

(iii) Find, in the general case, the inverse map (i.e. express $\mathbf{x}$ in terms of $\mathbf{x}^{\prime}$ in vector form).