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 ยท y. As we shall see later, this convention makes the transition from Euclidean to abstract spaces simpler and more natural.
It is only with the multiplication of vectors by one another that the geometry becomes richโJ. L. Synge
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ยฉ 1981 Plenum Press, New York
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Nowinski, J.L. (1981). Inner Product of Vectors. Norm. In: Applications of Functional Analysis in Engineering. Mathematical Concepts and Methods in Science and Engineering, vol 22. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-3926-7_3
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DOI: https://doi.org/10.1007/978-1-4684-3926-7_3
Publisher Name: Springer, Boston, MA
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