Quaternion multiplication seems suspiciously similar to the cross product. How is it derived?
Here is a description of the multiplication: Let $Q_1$ and $Q_2$ be two quaternions, which are defined, respectively, as $(w_1, x_1, y_1, z_1)$ and $(w_2, x_2, y_2, z_2)$.
$(Q_1 Q_2).w = (w_1w_2 - x_1x_2 - y_1y_2 - z_1z_2)$
$(Q_1 Q_2).x = (w_1x_2 + x_1w_2 + y_1z_2 - z_1y_2)$
$(Q_1 Q_2).y = (w_1y_2 - x_1z_2 + y_1w_2 + z_1x_2)$
$(Q_1 Q_2).z = (w_1z_2 + x_1y_2 - y_1x_2 + z_1w_2)$