Let \(p\) be an arbitrary quaternion and let \(q\) be a unit quaternion. Show that the real part of the product \(q p \bar{q}\) is equal to \(\Re(p)\). Show also that the imaginary part of the product \(q p \bar{q}\) is a \(3 \mathrm{D}\) rotation of \(\mathfrak{\Im}(p)\).