Scalars and Vectors - Vector Analysis for Computer Graphics - page 1

1 Scalars and Vectors
1.1 Introduction
Over thousands of years humans have developed various ways of counting things, whether they
were oxen, days, trees, or stars. Various symbols, or strings of symbols, such as 9 or IX, have
enabled us to count, record, compare, or communicate the number of things to our fellow
human beings—such quantities are called scalars .
Initially, the natural numbers (1, 2, 3, 4 ) were sufficient to count most things, but eventually
the class of integers emerged, which embraced the concepts of zero and negative numbers.
Today we employ integers to count populations, real numbers to record the temperature, and
transcendental numbers to express , e, and trigonometric functions.
Scalar quantities embrace most things in our everyday lives such as height, age, temperature,
page numbers, mass, distance, and tax rates. However, some things cannot be defined or
measured by a single number. For instance, velocity or force requires a magnitude and a
direction. So, too, does weight, which arises through one mass attracting another. In the case
of objects on the earth, weight is determined by the object's mass and the earth's mass and is
directed towards the earth's centre. The same object on the moon weighs less and is directed
towards the moon's centre.
A natural step would be to employ two or more scalars to record such quantities, which
is what happened about 150 years ago. After considerable debate, the term vector emerged to
describe quantities possessing magnitude and direction, and a simple, yet powerful algebra was
established to analyse and solve some challenging geometric problems.
1.2 Representing vector quantities
We can approach the subject of vectors from two directions: one employs an axiomatic basis;
the second is intuitive. The axiomatic approach begins by defining the meaning of a vector space
and the objects (vectors) that are members of this space, together with the axioms describing
their manipulation. The intuitive approach is much more visual and uses directed or oriented
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