The hydroxyl radical (OH) is one of the most interesting molecules in molecular dynamics. In particular, inelastic collisions of free radicals such as OH are profoundly important in environments ranging from combustion to astrochemistry. However, measuring the velocities of OH molecules in specific internal quantum states has proven to be very difficult. A method that can provide this important information is velocity-map imaging. Although this technique is very widely applicable in principle, it does require a sensitive and selective laser-ionization scheme. Here we show that, under the right conditions, velocity-map imaging can be applied to the study of the inelastic scattering of OH using crossed-molecular-beam methods. We measure fully quantum-state-specified product angular distributions for OH collisions with helium and argon. The agreement between exact close-coupling quantum scattering calculations on ab initio potential energy surfaces and experimental data is generally very satisfactory, except for scattering in the most forward directions.