Vectors - 3D Math Primer for Graphics and Game Development - page 34

Note that other authors may use different conventions. One common con-
vention used frequently when writing vectors by hand, is to draw an arrow
over the vector, for example, a.

Before we go any further, a bit of context is in order concerning the
perspective that we are adopting about vectors. The branch of mathemat-
ics that deals primarily with vectors and matrices is called linear algebra,
a subject that assumes the abstract definition given previously: a vector is
an array of numbers. This highly generalized approach allows for the explo-
ration of a large set of mathematical problems. In linear algebra, vectors
and matrices of dimension n are used to solve a system of n linear equations
for n unknowns, without knowing or caring what physical significance, if
any, is attached to any of the numbers. This is certainly a fascinating and
highly practical study, but it is not of primary interest to our investigation
of 3D math. For 3D math, we are mostly concerned with the geometric
interpretations of vectors and vector operations.
Our focus is geometric, so we omit many details and concepts of linear
algebra that do not further our understanding of 2D or 3D geometry. Even
though we occasionally discuss properties or operations for vectors of an
arbitrary dimension n, we will usually focus on 2D, 3D, and (later) 4D
vectors and matrices. Even when the numbers in a vector do not have any
physical significance, the ability to visualize the linear algebra operations
is of some utility, so learning how to interpret the operations geometrically
is useful even in nonphysical applications. Some more context about how
the topics in this topic fit into the bigger picture of linear algebra can be
found in Section 4.3.

2.2 Geometric Definition of Vector

Now that we have discussed what a vector is mathematically, let's look at a
more geometric interpretation of vectors. Geometrically speaking, a vector
is a directed line segment that has magnitude and direction.

• The magnitude of a vector is the length of the vector. A vector may

have any nonnegative length.

• The direction of a vector describes which way the vector is pointing in

space. Note that “direction” is not exactly the same as “orientation,”
a distinction we will reexamine in Section 8.1.