Graphics Reference
In-Depth Information
2.11 Perpendicular vectors
When working in 2D, we often employ vectors that are perpendicular to some reference vector.
For example, Fig. 2.31 shows two vectors
v
and
n
, where
n
is perpendicular to
v
and is expressed
mathematically as
n
⊥
v
.
n
v
Figure 2.31.
The symbol
⊥
(pronounced “perp”) can be regarded as an operator, such that given a
vector
v
,
v
⊥
is a vector perpendicular to
v
. However, there is a problem: which way does
v
⊥
point? Figure 2.32 shows the two possibilities. Either way is valid. However, mathe-
matics does employ a convention where a counter-clockwise rotation is positive, which
also supports the right-handed axial system used in this topic. Consequently, the orien-
tation shown in Fig. 2.32(a) is the one adopted. The next step is to find the components
of
v
⊥
.
v
⊥
v
⊥
v
v
(a)
(b)
Figure 2.32.
Figure 2.33 shows vector
v
=
a
i
+
b
j
, which makes an angle with the x-axis, and 90
−
with
the y-axis. If we transpose
v
's components to
a
j
, we create a second vector, which must
be perpendicular to
v
, because the angle between the two vectors is 90
. Therefore, we can state
that
−
b
i
+
=
+
v
a
i
b
j
v
⊥
=−
+
b
i
a
j
(2.22)
