Mathematical Methods for Physics and Engineering - Matematica.NET

7Vector algebraThis chapter introduces space vectors and their manipulation. Firstly we deal withthe description and algebra of vectors, then we consider how vectors may be usedto describe lines and planes and finally we look at the practical use of vectors infinding distances. Much use of vectors will be made in subsequent chapters; thischapter gives only some basic rules.7.1 Scalars and vectorsThe simplest kind of physical quantity is one that can be completely specified byits magnitude, a single number, together with the units in which it is measured.Such a quantity is called a scalar and examples include temperature, time anddensity.A vector is a quantity that requires both a magnitude (≥ 0) and a direction inspace to specify it completely; we may think of it as an arrow in space. A familiarexample is force, which has a magnitude (strength) measured in newtons and adirection of application. The large number of vectors that are used to describethe physical world include velocity, displacement, momentum and electric field.Vectors are also used to describe quantities such as angular momentum andsurface elements (a surface element has an area and a direction defined by thenormal to its tangent plane); in such cases their definitions may seem somewhatarbitrary (though in fact they are standard) and not as physically intuitive as forvectors such as force. A vector is denoted by bold type, the convention of thisbook, or by underlining, the latter being much used in handwritten work.This chapter considers basic vector algebra and illustrates just how powerfulvector analysis can be. All the techniques are presented for three-dimensionalspace but most can be readily extended to more dimensions.Throughout the book we will represent a vector in diagrams as a line togetherwith an arrowhead. We will make no distinction between an arrowhead at the212