Graphics Reference
In-Depth Information
Y
a
v
⊥
α
v
b
90
°
-
α
α
a
-
b
X
Figure 2.33.
Further confirmation is found in the dot product where
v
⊥
=
v
·
a
i
+
b
j
·
−
b
i
+
a
j
=−
ab
+
ab
=
0
A second way of demonstrating this is to convert
v
into a complex number:
i.e.,
a
i
+
b
j
≡
a
+
ib
(2.23)
where
√
i
=
−
1
If we multiply a
+
ib by i, it effectively rotates it through 90
:
i
2
b
ia
+
ib
=
ai
+
=−
b
+
ai
where
−
b
+
ia
≡−
b
i
+
a
j
Although the sign change and component switching take a simple operation, it can be repre-
sented formally by this determinant:
=−
i j
ab
v
⊥
=−
b
i
+
a
j
It may be obvio
us that the
magnitude
v
⊥
equals the magnitude
v
. If not, it can be shown
that
v
⊥
=
x
v
=
x
v
+
y
v
2
.
Now we already know that the dot product reveals the angle between two vectors:
−
+
y
v
=
v
a
·
b
=
a
b
cos
