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-alpha 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 + H-2 --> H + H2O: sigma(R)(0.17 eV) = (0.08 +/- 0.03) Angstrom(2), sigma(R)(0.22 eV) = (0.60 +/- 0.30) Angstrom(2); OH + D-2 --> D + HOD: sigma(R)(0.28 eV) = (0.22 +/- 0.05) Angstrom(2), sigma(R)(0.37 eV) = (0.43 +/- 0.009) Angstrom(2). 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: sigma(R)(1.5 eV) = (0.07 +/- 0.04) Angstrom(2), sigma(R)(1.8 eV) = (0.10 +/- 0.03) Angstrom(2), and sigma(R)(2.2 eV) = (0.11 +/- 0.03) Angstrom(2). 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 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 + H-2 and OH + D-2 than in the reaction H + D2O. This, together with a preferential population of the symmetric (II)-I-2(A') Lambda-doublet state (with the unpaired pi 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 ''latter'' position of the reaction coordinate.