In this work we present a method for calculating the stationary state wave functions and reaction probabilities of a multidimensional reactive scattering system. Our approach builds upon the counter-propagating wave methodology (CPWM) developed by Poirier and co-workers for calculating one-dimensional stationary state wave functions. The method involves the formulation of a bipolar decomposition for multidimensional stationary scattering wave functions within the context of a reaction path Hamiltonian, so we refer to this work as the bipolar reaction path Hamiltonian (BRPH) approach. Benchmark calculations are presented for several 2D model scattering systems with linear reaction coordinates. We show that the BRPH approach is competitive with conventional calculations based on discrete variable representation (DVR) methods.