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then
A
†
=
λ
+
v
−
B
−
T
.
Let's illustrate this reversion process with an example.
Given three vectors
a
=
2
e
1
+
3
e
2
b
=
4
e
1
−
2
e
2
c
=
e
1
+
e
2
the products
ab
and
ba
are
ab
=
(
2
e
1
+
3
e
2
)(
4
e
1
−
2
e
2
)
=
2
−
16
e
12
ba
=
(
4
e
1
−
2
e
2
)(
2
e
1
+
3
e
2
)
=
2
+
16
e
12
.
Thus
(
ab
)
†
=
ba
.
Furthermore, the products
abc
and
cba
are
abc
=
(
2
e
1
+
3
e
2
)(
4
e
1
−
2
e
2
)(
e
1
+
=−
14
e
1
+
e
2
)
18
e
2
cba
=
(
e
1
+
e
2
)(
4
e
1
−
2
e
2
)(
2
e
1
+
3
e
2
)
=−
14
e
1
+
18
e
2
.
And as there are only vectors terms there are no sign changes.
Reversion plays an important role in rotors and we will meet them again in the
next chapter.
6.19 Summary
This chapter has covered the basic ideas behind geometric algebra which offers
an algebraic framework for oriented lines (vectors), oriented planes (bivectors) and
oriented volumes (trivectors), not to mention higher dimensional objects. We have
yet to discover how they offer an alternative way of rotating points in the plane and
in 3D space.
6.19.1 Summary of Multivector Operations
Inner product:
2D vectors
a
=
a
1
e
1
+
a
2
e
2
b
=
b
1
e
1
+
b
2
e
2
a
·
b
=|
a
||
b
|
cos
β
=
a
1
b
1
+
a
2
b
2
.
