Convert the following expressions from suffix notation (assuming the summation convention in three dimensions) into standard notation using vectors and/or matrices, where possible, identifying the one expression that is incorrectly formed:

(i) $\delta_{i j}$,

(ii) $\delta_{i i} \delta_{i j}$,

(iii) $\delta_{l l} a_{i} b_{j} C_{i j} d_{k}-C_{i k} d_{i}$,

(iv) $\epsilon_{i j k} a_{k} b_{j}$,

(v) $\epsilon_{i j k} a_{j} a_{k}$.

Write the vector triple product $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ in suffix notation and derive an equivalent expression that utilises scalar products. Express the result both in suffix notation and in standard vector notation. Hence or otherwise determine $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ when $\mathbf{a}$ and $\mathbf{b}$ are orthogonal and $\mathbf{c}=\mathbf{a}+\mathbf{b}+\mathbf{a} \times \mathbf{b}$.