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7.7.2 Adding and Subtracting Quaternions
Given two quaternions q 1 and q 2 ,
q 1 =[ s 1 , v 1 ]=[ s 1 + x 1 i + y 1 j + z 1 k ]
q 2 =[ s 2 , v 2 ]=[ s 2 + x 2 i + y 2 j + z 2 k ]
(7.86)
they are equal if, and only if, their corresponding terms are equal. Further-
more, like vectors, they can be added and subtracted as follows:
q 1 ± q 2 =[( s 1 ±
s 2 )+( x 1 ±
x 2 ) i +( y 1 ±
y 2 ) j +( z 1 ±
z 2 ) k ]
(7.87)
7.7.3 Multiplying Quaternions
Hamilton discovered that special rules must be used when multiplying quater-
nions:
i 2 = j 2 = k 2 = ijk =
1
ij = k , jk = i , ki = j
ji =
k , kj =
i , ik =
j
(7.88)
Note that although quaternion addition is commutative, the rules make quater-
nion multiplication non-commutative.
Given two quaternions q 1 and q 2 ,
q 1 =[ s 1 , v 1 ]=[ s 1 + x 1 i + y 1 j + z 1 k ]
q 2 =[ s 2 , v 2 ]=[ s 2 + x 2 i + y 2 j + z 2 k ]
(7.89)
the product q 1 q 2 ,isgivenby
q 1 q 2 =[( s 1 s 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 )+( s 1 x 2 + s 2 x 1 + y 1 z 2 − y 2 z 1 ) i
+( s 1 y 2 + s 2 y 1 + z 1 x 2
z 2 x 1 ) j +( s 1 z 2 + s 2 z 1 + x 1 y 2
x 2 y 1 ) k
(7.90)
which can be rewritten using the dot and cross product notation as
q 1 q 2 =[( s 1 s 2
v 1 ·
v 2 ) ,s 1 v 2 + s 2 v 1 + v 1 ×
v 2 ]
(7.91)
7.7.4 The Inverse Quaternion
Given the quaternion
q =[ s + x i + y j + z k ]
(7.92)
the inverse quaternion q 1 is
q 1 = [ s
x i
y j
z k ]
(7.93)
| q |
2
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