The conjugate quaternion is \(\bar{q}=q_{0}-q_{1} \mathbf{i}-q_{2} \mathbf{j}-q_{3} \mathbf{k}\), so \(q=\bar{q}\) means that \(q\) is real, and \(q=-\bar{q}\) means that \(q\) is purely imaginary. Show that conjugate product \(\overline{p \bar{q}}\) is equal to the product of the conjugates \(q \bar{p}\) in reverse order, and

\[
|q|^{2}=q \bar{q}=\bar{q} q=q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2} .
\]