Gluck and Ziller proved that Hopf vector fields on $S^{3}$ have minimum volume among all unit vector fields. Thinking of $S^{3}$ as a Lie group, Hopf vector fields are exactly those with unit length which are left or right invariant, and $TS^{3}$ is a trivial vector bundle with a connection induced by the adjoint representation. We prove the analogue of the stated result of Gluck and Ziller for the representation given by quaternionic multiplication. The resulting vector bundle over $S^{3}$, with the Sasaki metric, has as well no parallel unit sections. We provide an application of a double point calibration, proving that the submanifolds determined by the left and right invariant sections minimize volume in their homology classes.