Graphics Reference
In-Depth Information
we extract the scalar term using:
2
+
3
e
1
+
2
e
2
−
5
e
12
+
6
e
123
0
=
2
the vector term using
2
+
3
e
1
+
2
e
2
−
5
e
12
+
6
e
123
1
=
3
e
1
+
2
e
2
the bivector term using:
2
+
3
e
1
+
2
e
2
−
5
e
12
+
6
e
123
2
=−
5
e
12
and the trivector term using:
2
+
3
e
1
+
2
e
2
−
5
e
12
+
6
e
123
3
=
6
e
123
.
It is also worth pointing out that the inner vector product converts two grade 1
elements, i.e. vectors, into a grade 0 element, i.e. a scalar, whereas the outer vector
product converts two grade 1 elements into a grade 2 element, i.e. a bivector. Thus
the inner product is a grade lowering operation, while the outer product is a grade
raising operation. These qualities of the inner and outer products are associated
with higher grade elements in the algebra. This is why the scalar product is renamed
as the inner product, because the scalar product is synonymous with transforming
vectors into scalars. Whereas, the inner product transforms two elements of grade
n
into a grade
n
−
1 element.
6.12 Redefining the Inner and Outer Products
As the geometric product is defined in terms of the inner and outer products, it
seems only natural to expect that similar functions exist relating the inner and outer
products in terms of the geometric product. Such functions do exist and emerge
when we combine the following two equations:
ab
=
a
·
b
+
a
∧
b
(6.10)
ba
=
a
·
b
−
a
∧
b
.
(6.11)
Adding and subtracting (
6.10
) and (
6.11
)wehave
1
2
(
ab
a
·
b
=
+
ba
)
(6.12)
1
2
(
ab
a
∧
b
=
−
ba
).
(6.13)
Equations (
6.12
) and (
6.13
) are used frequently to define the products between dif-
ferent grade elements.
