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the second part must be
a
⊥
=
ˆ
ˆ
n
(
n
∧
a
).
From Fig.
12.2
we see that
a
=
a
⊥
−
a
(12.3)
= ˆ
n
(
n
ˆ
∧
a
)
−
(
n
ˆ
·
a
)
n
.
ˆ
(12.4)
Equation (
12.4
) contains the product of a vector
n
and a bivector
ˆ
n
ˆ
∧ ˆ
a
which anti-
commute:
n
2
(
ˆ
n
(
ˆ
n
ˆ
∧
a
)
=
na
ˆ
−
a
n
)
ˆ
1
2
(
a
−
ˆ
ˆ
=
na
n
)
whereas,
1
2
(
na
(
n
∧
a
)
n
=
−
a n
)
n
1
2
(
ˆ
=
na
n
ˆ
−
a
)
therefore, we can write (
12.4
)as
a
=−
ˆ
·
ˆ
−
ˆ
∧
ˆ
(
n
a
)
n
(
n
a
)
n
which simplifies to
a
=−
(
n
ˆ
·
a
+ ˆ
n
∧
a
)
n
.
ˆ
(12.5)
By substituting (
12.1
)in(
12.5
)wehave
a
=−ˆ
na
n
ˆ
(12.6)
which is rather elegant!
To illustrate (
12.6
), consider the scenario shown in Fig.
12.3
where we see a
mirror placed on the
zx
-plane with normal vector
j
or
e
2
. The vector to be reflected
is
a
=
i
+
j
−
k
which can also be expressed as
a
=
e
1
+
e
2
−
e
3
.
Using (
12.6
)wehave
a
=−
e
3
)
e
2
which, using the rules of multivectors simplifies to
a
=−
e
2
(
e
1
+
e
2
−
e
2
e
1
e
2
−
e
2
e
2
e
2
+
e
2
e
3
e
2
=
e
1
−
e
2
−
e
3
k
and is confirmed by Fig.
12.3
. Now let's see how these ideas can be generalised into
3D rotations.
=
i
−
j
−
