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and substitute it in (
12.12
):
R
θ
=
b
n
=
b
ˆ
+
b
ˆ
a
+
b
|ˆ
a
|
+
b
1
a
ˆ
=
.
+
b
|ˆ
a
|
Similarly,
R
†
n b
θ
= ˆ
+
b
a
ˆ
b
=
+
b
|ˆ
a
|
a b
1
+ ˆ
=
.
+
b
|
ˆ
a
|
It is possible to show that
2
(
1
+
b
·
b
)
|ˆ
a
|=
+ ˆ
a
which permits us to propose an alternative solution
+
b
ˆ
1
a
R
θ
=
2
(
1
(12.14)
·
b
)
+ ˆ
a
a b
1
+ ˆ
R
θ
=
2
(
1
.
(12.15)
·
b
)
+ ˆ
a
Now let's put these definitions to the test.
Figure
12.9
shows vector
a
aligned with the
e
1
axis and
b
aligned with the
e
2
axis. Therefore, the rotor is acting in the
e
12
plane with an angle of 45° to effect a
rotation of 90°. Using our knowledge of rotors, it is obvious that
ˆ
Fig. 12.9
Vector
a
rotates
ˆ
to
b
