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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

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