Chapter 10 - The Vector Space Model: The Algebra of Directions

Abstract

There are n independent 1-D directions in an n-dimensional physical space, and they can conveniently be drawn as vectors at the origin. Mathematically, they form a vector space Rn that can be added and scaled with real numbers to produce other legitimate 1-D directions. The metric of the directions in the physical space (typically Euclidean) can be used to induce a metric in this mathematical representation. That gives a model of the directions in physical space in terms of the geometric algebra of a metric vector space Rn. The vector space model thus constructed is indeed a good computational representation of spatial directions at the origin. The vector space model is the natural model to treat angular relationships at a single location. In a 3-D Euclidean space, geometrical directions can be indicated by vectors or bivectors (which are always 2-blades). The scalars and trivectors have trivial directional aspects and are mostly used for their orientations and magnitudes. Relative angles between the directional elements are fully represented by their geometric ratios.