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In-Depth Information
q 1 q 2 = cos 15°
sin 15° j cos 30°
sin 30° j
+
+
=
cos 15° cos 30°
sin 15° sin 30°
+
cos 15° sin 30° j
+
cos 30° sin 15° j
2
2 +
2
2
=
j
which is a quaternion that rotates 90° about the y -axis. Using the matrix ( 11.4 )we
have
001
010
100
which rotates points about the y -axis by 90°.
11.5 Eigenvalue and Eigenvector
Although there is no doubt that ( 11.3 ) is a rotation matrix, we can secure further
evidence by calculating its eigenvalue and eigenvector. The eigenvalue should be θ ,
where
Tr qpq 1 =
1
+
2 cos θ.
The trace of ( 11.3 )is
Tr qpq 1 =
2 s 2
+ x 2
2 s 2
+ y 2
2 s 2
+ z 2
1
+
1
+
1
2 s 2
z 2
4 s 2
x 2
y 2
=
+
+
+
+
3
4 s 2
=
1
4 cos 2 (θ/ 2 )
=
1
4sin 2 (θ/ 2 )
=
+
4 cos θ
1
=
4 cos θ
+
2
2 cos θ
1
=
1
+
2 cos θ
and
2 Tr qpq 1
1 .
1
cos θ =
To compute the eigenvector of ( 11.3 ) we use the three equations derived in Chap. 9:
v 1 =
(a 22
1 )(a 33
1 )
a 23 a 32
v 2 =
(a 33
1 )(a 11
1 )
a 31 a 13
v 3 = (a 11
1 )(a 22
1 ) a 12 a 21 .
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