Graphics Reference
In-Depth Information
Axiom 6.7
a
2
2
.
=|
a
|
6.10 Notation
Having abandoned
i
,
j
,
k
for
e
1
,
e
2
,
e
3
, it is convenient to convert geometric products
e
1
e
2
···
e
n
to
e
12
···
n
. For example,
e
1
e
2
e
3
≡
e
123
. Furthermore, we must get used to
the following substitutions:
e
i
e
i
e
j
=
e
j
e
21
=−
e
12
e
312
=
e
123
e
112
=
e
2
e
121
=−
e
2
.
6.11 Grades, Pseudoscalars and Multivectors
As geometric algebra embraces such a wide range of objects, it is convenient to
grade
them as follows: scalars are grade 0, vectors are grade 1, bivectors are grade 2,
and trivectors are grade 3, and so on for higher dimensions. In such a graded algebra
it is traditional to call the highest grade element a
pseudoscalar
. Thus in 2D the
pseudoscalar is
e
12
and in 3D the pseudoscalar is
e
123
.
One very powerful feature of geometric algebra is the idea of a
multivector
,
which is a linear combination of a scalar, vector, bivector, trivector, or any other
higher dimensional object. For example the following are multivectors:
A
=
3
+
(
2
e
1
+
3
e
2
+
4
e
3
)
+
(
5
e
12
+
6
e
23
+
7
e
31
)
+
8
e
123
B
=
2
+
(
2
e
1
+
2
e
2
+
3
e
3
)
+
(
4
e
12
+
5
e
23
+
6
e
31
)
+
7
e
123
and we can form their sum:
A
+
B
=
5
+
(
4
e
1
+
5
e
2
+
7
e
3
)
+
(
9
e
12
+
11
e
23
+
13
e
31
)
+
15
e
123
or their difference:
A
−
B
=
1
+
(
e
2
+
e
3
)
+
(
e
12
+
e
23
+
e
31
)
+
e
123
.
We can even form their product
AB
.
We can isolate any grade of a multivector using the following notation:
g
where
g
identifies a particular grade. For example, say we have the following mul-
tivector:
multivector
2
+
3
e
1
+
2
e
2
−
5
e
12
+
6
e
123
