Graphics Reference
In-Depth Information
6.5 Inner and Outer Products
Let's assume that the products
ij
and
ji
in (
6.1
) anticommute:
ji
=−
ij
. Therefore,
a
1
b
1
i
2
a
2
b
2
j
2
ab
=
+
+
(a
1
b
2
−
a
2
b
1
)
ij
(6.2)
and if we reverse the product to
ba
we obtain
a
1
b
1
i
2
a
2
b
2
j
2
a
2
b
1
)
ij
.
(6.3)
From (
6.2
) and (
6.3
) we see that the product of two vectors contains a symmetric
component
ba
=
+
−
(a
1
b
2
−
a
1
b
1
i
2
+
a
2
b
2
j
2
and an antisymmetric component
a
2
b
1
)
ij
.
Geometric algebra defines the product
ab
as the sum of two other products called
the
inner
and
outer
products. The inner product has the form
(a
1
b
2
−
·
=|
||
|
a
b
a
b
cos
β
(6.4)
where
a
·
b
=
(a
1
i
+
a
2
j
)
·
(b
1
i
+
b
2
j
)
=
a
1
b
1
i
·
i
+
a
1
b
2
i
·
j
+
a
2
b
1
j
·
i
+
a
2
b
2
j
·
j
a
2
b
2
which is the familiar scalar product. The outer product uses the wedge '
=
a
1
b
1
+
∧
' symbol,
which is why it is also called the
wedge product
and has the form
a
∧
b
=|
a
||
b
|
sin
β
i
∧
j
(6.5)
where
∧
=
+
∧
+
a
b
(a
1
i
a
2
j
)
(b
1
i
b
2
j
)
=
a
1
b
1
i
∧
i
+
a
1
b
2
i
∧
j
+
a
2
b
1
j
∧
i
+
a
2
b
2
j
∧
j
=
(a
1
b
2
−
a
2
b
1
)
i
∧
j
which enables us to write
ab
=
a
·
b
+
a
∧
b
(6.6)
ab
=|
a
||
b
|
cos
β
+|
a
||
b
|
sin
β
i
∧
j
.
(6.7)
6.6 The Geometric Product in 2D
Clifford named the sum of the two products the
geometric product
, which means
that (
6.6
) reads: The geometric product
ab
is the sum of the inner product “
a
dot
b
”
and the outer product “
a
wedge
b
”.
