Graphics Reference
In-Depth Information
As we can see from Eqs. (7.17) and (7.18)
q
1
q
2
=
q
1
q
2
But if we reverse q
1
and q
2
and expand q
2
q
1
, we find that
q
2
q
1
=
s
2
−
v
2
s
1
−
v
1
q
2
q
1
=
s
1
s
2
−
v
1
·
v
2
−
s
1
v
2
−
s
2
v
1
−
v
1
×
v
2
which proves that
q
1
q
2
=
q
2
q
1
(7.19)
Now let's prove that
q
¯
q
=¯
qq
Starting with
q
q
¯
=
s
+
v
s
−
v
2
q
¯
q
=
s
2
−
v
·
−
v
−
v
×
v
=
s
2
+
v
Similarly,
¯
qq
=
s
−
v
s
+
v
2
s
2
s
2
¯
qq
=
−
−
v
·
v
−
v
×
v
=
+
v
Therefore,
q
¯
q
=¯
qq
(7.20)
7.7 The norm of a quaternion
The
absolute value
or
magnitude
of a complex number
+
a
b
i
is defined as
√
a
2
=
+
z
b
2
Similarly, the
norm
or
magnitude
of a quaternion
q
=
s
+
x
i
+
y
j
+
z
k
