In our study of computer graphics we will frequently

In our study of computer graphics we will frequently use vectors to solve a va-
riety of problems in such diverse areas as geometric modelling, transformations,
projections, visibility determination, lighting, shading and texturing, and the de-
velopment of curves, surfaces and deformations. Thus it is important to gain a
thorough understanding of vector algebra.
2.1 Some Basic Definitions and Notation
Let us start by defining the terms scalar and vector . A scalar is a quantity that is
completely determined by a single numerical value, which consists of a possibly
signed magnitude (i.e. a real number). For instance, quantities such as tempera-
ture, time, length, mass, and speed are all represented by scalars. A vector, on the
other hand, is a quantity that is determined by a direction and a magnitude . It is
a directed quantity represented by an arrow. The direction and the length of this
arrow determine the vector’s direction and magnitude, respectively.
In everyday language it is common to use quantities such as speed and velocity
interchangeably. This use however is inaccurate as the speed of a vehicle repre-
sents the magnitude of its movement alone and gives no indication of its direction
of movement, while the velocity of a vehicle represents both its magnitude and its
direction of movement. A scalar is a single dimensional quantity, while a vector
is a multidimensional quantity.
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66 Mathematical and Computer Programming Techniques for Computer Graphics
In this book we denote vectors in bold italic notation such as v , v or v , while
scalars are denoted by Greek letters or non-bold italic characters such as λ or l .
A three-dimensional geometric vector can be seen as a translation in three-
dimensional Euclidean space E
3
. Given a point P in E
3
, we may use a vector
v to move this point to a new position P , as shown in Fig. 2.1 . Sometimes we
call the vector v a displacement vector , as it displaces point P to its new position
P . The distance between points P and P is called the magnitude of the vector v
and it is denoted by | v | or the non-bold italic version of the vector name v . This
distance is the same for all points P in E
3
. Thus, the original position of point P
is immaterial. For all vectors v we say that
|
v
| ≥
0.
There exists a special vector whose magnitude is zero. This vector translates
every point P onto itself, i.e. the position of the point is left unchanged and the
magnitude of the vector is zero. We call this vector the zero vector or null vector
and we denote it by 0 or 0 when there is no notational ambiguity (i.e. when it can
not be misinterpreted as a scalar). No direction is associated with the zero vector.
Thus,
0 = 0 (2.1)
It follows that
| v | > 0 for every v = 0 (2.2)
A vector v whose magnitude is equal to one (i.e. | v | = 1) is called a unit vector .