Multiphase motor drives are multi-dimensional systems, so multiphase vector control and SVPWM must be implemented in a multi-dimensional vector space. In this paper, multi-dimensional multiphase vector control is developed based on a novel multiphase SVPWM strategy, which could synthesize the required voltage vectors not only in d-q subspace but also in other subspaces. Therefore, all voltage and current harmonics, which contribute to the torque positively, are controllable. Throughout this paper, a five-phase permanent magnet motor drive system is developed as an example, and simulation and experimental results verify these proposed strategies. Moreover, these strategies can be easily extended to other multiphase drive systems with any number of phases.