In this paper we introduce Multinomial Squared Direction Cosines Regression as an alternative Multinomial Response Model. The proposed model offers an intuitive geometric interpretation to the task of estimating posterior class probabilities in multi-class problems. In specific, the latter probabilities correspond to the squared direction cosines between a given pattern and representative class exemplars that form a basis in the decision space. We demonstrate that the model allows for efficient training via a trust region based Newton's Method, provided that the number of model parameters is not too large. Experimental results on several benchmark classification problems show that it compares competitively against Logistic Regression Classifiers, Support Vector Machines, and Classification and Regression Tree models.