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and define
u
=-
nn
i
1
.
j
2
Clearly,
n
is orthogonal to
u
. We have
()
=
2
2
s
q
u
cos
q
u
-
cos
q
sin
q
un
+
sin
q
cos
q
nu
-
sin
q
nun
.
(20.6)
But by Proposition 20.2.6(2)
nu nununu
=- ∑
+
¥
=
¥
and
un ununun
=- ∑
+
¥
=
¥ .
Therefore,
n u
-=¥.
u n
2n
u
(20.7)
We also have
(
)
∑+ ¥
(
)
¥= ¥
(
)
¥=
nun nunnunnunn
=-
¥
(20.8)
(the last equality is Exercise 20.2.7). If we substitute equations (20.7) and (20.8) into
equation (20.6), we get
(
)
()
=
2
2
+
(
)
s
q
u
cos
q
-
sin
q
u
2
sin
q
cos
q
n
¥
u
(20.9)
=
cos
2
q
u
+
sin
2
q
n
¥
u
.
Since
u
,
n
¥
u
, and
n
are an orthonormal basis for
R
3
and equation (20.9) is just the
equation of a rotation in the plane through the origin with basis
u
and
n
¥
u
, s
q
must
be the map we claimed it was.
Finally, to prove (3) note that the rows of M
q
are just s
q
(
i
), s
q
(
j
), and s
q
(
k
). These
values are easily computed and shown to be as indicated. This finishes the proof of
the proposition.
Now the principal axis theorem (Theorem 2.5.5 in [AgoM05]) implies that every
rotation in
R
3
that fixes the origin is a rotation about some line through the origin.
Since that line has two unit direction vectors (one is the negative of the other), Propo-
sition 20.3.5(2) implies the following converse:
20.3.6 Proposition.
Every rotation R of
R
3
that fixes the origin is of the form s
q
for some non-zero quaternion
q
. In fact, we may assume that
q
is a unit quaternion
that is unique up to sign.
Proof.
Let R be the rotation through an angle q about the directed line through the
origin with unit direction vector
n
. If
