7

Vector algebra

This chapter introduces space vectors and their manipulation. Firstly we deal with

the description and algebra of vectors, then we consider how vectors may be used

to describe lines and planes and finally we look at the practical use of vectors in

finding distances. Much use of vectors will be made in subsequent chapters; this

chapter gives only some basic rules.

7.1 Scalars and vectors

The simplest kind of physical quantity is one that can be completely specified by

its magnitude, a single number, together with the units in which it is measured.

Such a quantity is called ascalarand examples include temperature, time and

density.

Avectoris a quantity that requires both a magnitude (≥0) and a direction in

space to specify it completely; we may think of it as an arrow in space. A familiar

example is force, which has a magnitude (strength) measured in newtons and a

direction of application. The large number of vectors that are used to describe

the physical world include velocity, displacement, momentum and electric field.

Vectors are also used to describe quantities such as angular momentum and

surface elements (a surface element has an area and a direction defined by the

normal to its tangent plane); in such cases their definitions may seem somewhat

arbitrary (though in fact they are standard) and not as physically intuitive as for

vectors such as force. A vector is denoted by bold type, the convention of this

book, or by underlining, the latter being much used in handwritten work.

This chapter considers basic vector algebra and illustrates just how powerful

vector analysis can be. All the techniques are presented for three-dimensional

space but most can be readily extended to more dimensions.

Throughout the book we will represent a vector in diagrams as a line together

with an arrowhead. We will make no distinction between an arrowhead at the