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We present a novel method for studying the shape of the Z boson transverse momentum distribution, Q_T, at hadron colliders in ppbar/pp -> Z/gamma* -> l^+l^-. The Q_T is decomposed into two orthogonal components; one transverse and the other parallel to the di-lepton thrust axis. We show that the transverse component is almost insensitive to the momentum resolution of the individual leptons and is thus more precisely determined on an event-by-event basis than the Q_T. Furthermore, we demonstrate that a measurement of the distribution of this transverse component is substantially less sensitive to the dominant experimental systematics (resolution unfolding and Q_T dependence of event selection efficiencies) reported in previous measurements of the Q_T distribution.