In this chapter we ease the transition from vectors in the plane to three dimensions and n-space. The angle between two vectors is often replaced by their scalar product, which is in many ways easier to work with and has special properties. Other kinds of vector product are useful too in geometry. An important issue for a set of vectors is whether it is dependent (i.e. whether one vector is a linear combination of the others). This apparently simple idea will have many ramifications in practical application.

We introduce the first properties of matrices, an invaluable handle on transformations in 2-, 3- and n-space. At this stage, besides identifying isometries with orthogonal matrices, we characterise the matrices of projection mappings, preparatory to the Singular Value Decomposition of Chapter 8 (itself leading to an optimal transform in Chapter 10.)

Vectors and handedness

This section is something like an appendix. The reader may wish to scan quickly through or refer back to it later for various formulae and notations. We reviewed vectors in the plane in Section 1.2.1. Soon we will see how the vector properties of having direction and length are even more useful in 3-space. The results of Section 1.2.1 still hold, but vectors now have three components rather than two.

Recapitulation – vectors

A vector ν consists of a magnitude |ν|, also called the length of ν, and a direction. Thus, as illustrated in Figure 7.1, ν is representable by any directed line segment AB with the same length and direction.