In this paper, we propose an interference alignment scheme based on the random vector quantization codebooks for the constant MIMO interference channel. We quantify an upper bound of the rate loss of the proposed scheme and derive the scaling law of the feedback load to maintain the constant rate loss relative to the interference alignment with perfect channel knowledge. Moreover, we compare the proposed scheme with the interference alignment scheme using analog feedback and show the advantage of the proposed scheme under the constraint of the feedback channel uses. From simulation results, we demonstrate that the proposed scheme can provide the constant rate loss by imposing the feedback load according to the derived scaling law.