Parallel and oblique proton fire hose instabilities are investigated in the presence of a small abundance of alpha particles with a nonzero drift velocity with respect to protons. Both instabilities scatter protons and alpha particles in the perpendicular direction with respect to the ambient magnetic field and decelerate both species with respect each other; especially the oblique fire hose effectively diffuses ions owing to its non-quasi-linear evolution. Linear Vlasov theory predicts that the presence of the alpha/proton drift enhances the maximum growth rates of the two instabilities in a similar way and that the parallel fire hose is typically the dominant instability. On the other hand, the oblique fire hose has often the maximum growth rate comparable to that of the parallel one and hybrid simulations show that the oblique instability may be active even when the parallel one is marginally stable. Consequently, both instabilities are relevant in the solar wind context.