Graphics Reference
In-Depth Information
These definitions are true, but they hide the agony that went on behind the scenes at the time.
Taking Eq. (2.11) as our definition, let's consider the orientation of
n
.
As we are employing a right-handed system of axes, our right hand also determines the
orientation of
n
. Therefore, with reference to our right hand, the thumb represents
a
, the first
finger represents
b
, and the middle finger provides the direction of
n
. It is convenient to think
of this operation as a rotation from
a
to
b
producing the perpendicular vector
n
. This is shown
in Fig. 2.20. Note that the vector product is very sensitive to the order of the two vectors. Switch
the vectors around and we get
b
×
a
=−
n
=−
b
a
sin
n
ˆ
and we see that the vector product breaks the commutative rule we take for granted with scalars.
In fact,
a
×
b
=−
b
×
a
.
b
n
a
θ
Figure 2.20.
Equation (2.11) not only determines the length of
n
, but also says something about the space
enclosed by
a
and
b
. And if we remove
ˆ
n
from the right-hand side, we obtain
a
b
sin,
which may remind you of another formula:
2
ab sin, which is the area of a triangle with sides
a and b with an enclosed angle . Figure 2.21 illustrates this.
1
C
D
b
θ
a
A
B
Figure 2.21.
The area of ABC equals
2
ab sin, which means that ab sin equals the area of the parallelogram
ABDC. Thus, when we compute the cross product
a
×
b
, the perpendicular vector
n
has a
length equal to the area of the parallelogram with sides
a
and
b
. Figure 2.22 illustrates
this relationship.
