Prove that $\left( \vec{a} \cdot \hat{i} \right) \left( \vec{a} \times \hat{i} \right) + \left( \vec{a} \cdot \hat{j} \right) \left( \vec{a} \times \hat{j} \right) + \left( \vec{a} \cdot \hat{k} \right) \left( \vec{a} \times \hat{k} \right) = \vec{0}$