A classification trees is one method of generating test cases from specifications. It partitions the input domain into a number of classifications. A classification tree is created to depict the relationship among the classifications. Test cases are then derived from this tree. However, classification trees have a number of shortcomings. Their expressive power is limited by the tree structure. They also rely heavily on human decisions in selecting test cases. This paper introduces an alternative approach to the generation of test cases. We first define classifications and classes formally. Then, we analyze the relations among classes and classifications and express these relations as class vectors. Test cases are then derived from the Cartesian product of vector partitions. The expressive power of vectors is better than that of a tree structure, and hence can be applied in some systems where test cases are complicated. Furthermore, more of the information obtained in the specifications is utilized. Hence, the amount of human decision-making is minimized.