(1) Write the vector equation \[ \vec{a} \times \vec{b} + (\vec{a} \cdot \vec{d}) \vec{c} = \vec{e} \] in index notation.

(2) Translate the index notation equation \[ \delta_{ij} c_j + \epsilon_{kji} a_k b_j = d_l \, e_m \, c_i \, b_l \, c_m \] into ordinary vector notation.

(3) Prove the following relationships

\[ \delta_{ij} \delta_{ij} = 3,\quad \varepsilon_{pqr} \varepsilon_{pqr} = 6,\quad \text{and} \quad \varepsilon_{pqi} \varepsilon_{pqj} = 2\delta_{ij} \]