Graphics Reference
In-Depth Information
Y
y
v
x
z
X
Z
Figure 2.3.
So far our definition of a vector accommodates a 2D line segment but also embraces three-
dimensional line segments. For example, Fig. 2.3 shows a 3D line segment with its Cartesian
components, where the vector
v
is represented by
⎡
⎤
x
y
z
⎣
⎦
v
=
Thus, a 2D vector has two components and a 3D vector has three components, which comprise
an
ordered triple
of its Cartesian components.
A vector's orientation and length are determined completely by the sign and value of its
Cartesian components. Fortunately, the length of any line segment is readily revealed by the
theorem of Pythagoras; but we also require a way of expressing this symbolically.
2.3 Vector notation
Boldface type is used for vector names. This is a universal convention and helps distinguish
scalars xyzrsst from vectors
a
b
c
d
n
p
q
. However, when we are dealing
with a line segment formed from two points, such as A and B, the associated vector is annotated
as
−
AB or
AB
, which represents a vector with its tail at A and head at B.
2.4 Length of a vector
We know that scalars can be positive or negative, and when we are only interested in the
absolute value of a scalar s (i.e., sign independent), we use the notation
, which effectively
strips away its sign. The same notation is used to represent the length of a vector, i.e.,
s
a
.It
is also possible to use
−
AB
or
AB
. Just to confuse matters, double vertical lines can be used
