We have studied the reaction of translationally excited hydroxyl radicals with molecular hydrogen at different center-of-mass energies. H/D atoms produced in the reaction have been detected under single collision conditions by means of vacuum ultraviolet laser-induced fluorescence at the Lyman-α transition. By calibrating the H/D signals from the reaction against H atom signals from the H2S and HCl photolysis, respectively, the following absolute reactive cross sections were determined: OH + H2 → H + H2O: σR(0.17 eV) = (0.08 ± 0.03) Å2, σR(0.22 eV) = (0.60 ± 0.30) Å2; OH + D2 → D + HOD: σR(0.28 eV) = (0.22 ± 0.05) Å, σR(0.37 eV) = (0.43 ± 0.09) Å2. With translationally excited H atoms, we have also studied the reverse reaction system, H + D2O. In this case the OD product radicals were detected under single collision conditions with quantum state resolution by means of laser-induced fluorescence. By calibrating the OD signals from the reaction against OH signals from the H2O2 photolysis, absolute reaction cross sections were measured for H + D2O → OD + HD: σR(1.5 eV) = (0.07 ± 0.04) Å2, σR(1.8 eV) = (0.10 ± 0.03) Å2, and σR(2.2 eV) = (0.11 ± 0.03) Å2. At different center-of-mass collision energies nascent population distributions of the OD product fine-structure components were determined. It has been found that at all collision energies OD radicals are produced exclusively in their vibrational ground state, with only a small amount of the total available energy appearing in the rotational degree of freedom. A comparison of the dependence of the reaction cross section on the translational energy shows that relative reagent translation is more effective to promote reactivity in the reaction OH + H2 and OH + D2 than in the reaction H + D2O. This, together with a preferential population of the symmetric 2Π(A′) Λ-doublet state (with the unpaired π orbital lying in the plane of rotation) at high OD rotational quantum numbers, suggests that the reaction H + D2O → OD + HD proceeds through a planar transition state via a direct mechanism, where the OD moiety acts as a spectator, and where the reaction barrier is located at a "later" position of the reaction coordinate.