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It should be no surprise that post-multiplying a vector p by the product mn rotates
it β .
The above results are summarised as follows:
nmp
=
pmn
rotates p
pnm
=
mnp
rotates p ,
β.
7.5.3 Rotate a Point About the Origin
In Chap. 9 on 3D rotations we show the origins of rotors in geometric algebra using
double reflections. The plane containing the vectors n and m is defined by the wedge
product m
n , which means we can write the product mn as
mn
=
m
·
n
+
m
n
and the product nm in the same plane as
nm
=
m
·
n
m
n
which accounts for the negative sign in the following bivector term
sin β e 12 ) .
Furthermore, if we make n and m unit vectors we can replace them by a rotor R β
whose magnitude is 1 because
nm
=|
n
||
m
|
( cos β
cos 2 β
sin 2 β
|
R β |=
+
=
1
therefore no scaling occurs, which means that
R β p
rotates p
i.e. anticlockwise, and
pR β
rotates p , β
i.e. clockwise .
The effect of this rotor is illustrated as follows:
=
e 1 +
p
e 2
R 45 ° =
cos 45°
sin 45° e 12
2
2
2
2
=
e 12
2
2
2
2
e 12 ( e 1 +
p =
R 45 ° p
=
e 2 )
2
2
2
2
2
2
2
2
=
e 1 +
e 2 +
e 2
e 1
2 e 2
=
asshowninFig. 7.7 .
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