Locative prepositions are probably one of the best studied lexical classes in linguistic semantics. As a result an impressive body of knowledge about their meaning and use has been accumulated in the past quarter of a century. 1 However, in comparison with the semantic literature about quantifiers, articles, and pronouns (to name some other well-studied parts of speech), a general formal framework for the representation of this knowledge is missing.2 Most of the theoretical results are either informal or formulated in terms of unanalyzed semantic primitives (like 'interior' for in and 'proximate' for near). The purpose of this paper is to provide a framework for the study of locative prepositions, based on the mathematical notion of a vector, i.e. a directed line-segment pointing from one point in space to another. Section 1 of this paper gives the basic outlines of such a vector-based semantics, section 2 provides the general mathematical background, and section 3 shows how semantic definitions for individual locative prepositions can be given in a straightforward way. The fruitfulness of the vector-based approach is demonstrated in section 4, where a number of general, algebraic properties of regions are formulated that explain constraints on modification and allow the postulation of universals of prepositions.