Signaling game problems investigate communication scenarios where encoder(s) and decoder(s) have misaligned objectives due to the fact that they either employ different cost functions or have inconsistent priors. This problem has been studied in the literature for scalar sources under various setups. In this paper, we consider multi-dimensional sources under quadratic criteria in the presence of a bias leading to a mismatch in the criteria, where we show that the generalization from the scalar setup is more than technical. We show that the Nash equilibrium solutions lead to structural richness due to the subtle geometric analysis the problem entails, with consequences in both system design, the presence of linear Nash equilibria, and an information theoretic problem formulation. We first provide a set of geometric conditions that must be satisfied in equilibrium considering any multi-dimensional source. Then, we consider independent and identically distributed sources and characterize necessary and sufficient conditions under which an informative linear Nash equilibrium exists. These conditions involve the bias vector that leads to misaligned costs. Depending on certain conditions related to the bias vector, the existence of linear Nash equilibria requires sources with a Gaussian or a symmetric density. Moreover, in the case of Gaussian sources, our results have a rate-distortion theoretic implication that achievable rates and distortions in the considered game theoretic setup can be obtained from its team theoretic counterpart.