(i) The spherical polar unit basis vectors $\mathbf{e}_{r}, \mathbf{e}_{\phi}$ and $\mathbf{e}_{\theta}$ in $\mathbb{R}^{3}$ are given in terms of the Cartesian unit basis vectors $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ by

$$\begin{aligned} &\mathbf{e}_{r}=\mathbf{i} \cos \phi \sin \theta+\mathbf{j} \sin \phi \sin \theta+\mathbf{k} \cos \theta \\ &\mathbf{e}_{\theta}=\mathbf{i} \cos \phi \cos \theta+\mathbf{j} \sin \phi \cos \theta-\mathbf{k} \sin \theta \\ &\mathbf{e}_{\phi}=-\mathbf{i} \sin \phi+\mathbf{j} \cos \phi \end{aligned}$$

Express $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ in terms of $\mathbf{e}_{r}, \mathbf{e}_{\phi}$ and $\mathbf{e}_{\theta}$.

(ii) Use suffix notation to prove the following identity for the vectors $\mathbf{A}, \mathbf{B}$, and $\mathbf{C}$ in $\mathbb{R}^{3}$ :

$$(\mathbf{A} \times \mathbf{B}) \times(\mathbf{A} \times \mathbf{C})=(\mathbf{A} \cdot \mathbf{B} \times \mathbf{C}) \mathbf{A}$$