Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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

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