Scalar Products and Euclidean Distances

Abstract

Scalar products are functions that are closely related to Euclidean distances. They are often used as an index for the similarity of a pair of vectors. A particularly well-known variant is the product-moment correlation for (deviation) scores. Scalar products have convenient mathematical properties and, thus, it seems natural to ask whether they can serve not only as indexes but as models for judgments of similarity. Although there is no direct way to collect scalar product judgments, it seems possible to derive scalar products from “containment” questions such as “How much of A is contained in B?” Since distance judgments can be collected directly, but scalar products are easier to handle numerically, it is also interesting to study whether distances can be converted into scalar products.