Scalars and vectors

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 a scalar and examples include temperature, time and
density.
A vector is 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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