Inner Product of Vectors. Norm

Abstract

The incorporation of a metric into the axiomatic system of affine spaces depends on the method chosen for evaluating the magnitude of arbitrarily directed vectors. The procedure universally adopted was probably suggested by Grassman and Gibbs. It consists of deriving the notion of length, or norm, of vectors from the concept of the inner (also called scalar, or dot) product of vectors. We shall subsequently denote the inner product of two vectors, x and y say, by the symbol (x, y) instead of the usual notation x · y. As we shall see later, this convention makes the transition from Euclidean to abstract spaces simpler and more natural.