Vectors - 3D Math Primer for Graphics and Game Development - page 53

which is the same as Equation (2.3). The proof of the magnitude equation
in 3D is only slightly more complicated.
For any positive magnitude m, there are an infinite number of vectors
of magnitude m. Since these vectors all have the same length but different
directions, they form a circle when the tails are placed at the origin, as
shown in Figure 2.15.

Figure 2.15
For any positive magnitude,
there are an infinite number of
vectors with that magnitude

2.9 Unit Vectors

For many vector quantities, we are concerned only with direction and not
magnitude: “Which way am I facing?” “Which way is the surface ori-
ented?” In these cases, it is often convenient to use unit vectors. A unit
vector is a vector that has a magnitude of one. Unit vectors are also known
as normalized vectors.
Unit vectors are also sometimes simply called normals; however, a warn-
ing is in order concerning terminology. The word “normal” carries with it
the connotation of “perpendicular.” When most people speak of a “nor-
mal” vector, they are usually referring to a vector that is perpendicular to
something. For example, a surface normal at a given point on an object
is a vector that is perpendicular to the surface at that location. However,
since the concept of perpendicular is related only to the direction of a vec-
tor and not its magnitude, in most cases you will find that unit vectors are
used for normals instead of a vector of arbitrary length. When this topic
refers to a vector as a “normal,” it means “a unit vector perpendicular