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2
2
qqqqqq q
==⋅
=
=
q
.
(4.36)
This leads us to a formula for the multiplicative inverse of a quaternion.
1
Theorem 4.5. The inverse of a nonzero quaternion q , denoted by
q
, is given
by
q
q
1
=
.
(4.37)
2
q
Proof. Applying Equation (4.36), we have
2
qq
q
qq
1
qq
===
1
(4.38)
2
2
and
2
qq
q
qq
1
q
q
===
1
,
(4.39)
2
2
thus proving the theorem.
4.6.2 Rotations with Quaternions
A rotation in three dimensions can be thought of as a function φ that maps
onto itself. For φ to represent a rotation, it must preserve lengths, angles, and
handedness. Length preservation is satisfied if
()
φ
PP .
=
(4.40)
The angle between the line segments connecting the origin to any two points P
and P is preserved if
()( )
φ φ
PPP
=
.
(4.41)
1
2
1
2
Finally, handedness is preserved if
() ( ) (
)
φ
P
×
φ
P
=
φ
P
×
P
.
(4.42)
1
2
1
2
Extending the function φ to a mapping from  onto itself by requiring that
(
)
(
)
φ s
+=+
v
s φ
v allows us to rewrite Equation (4.41) as