Graphics Reference
In-Depth Information
which is another quaternion. You may wish to evaluate
q
2
q
1
and show that
q
1
q
2
=
q
2
q
1
.
5.5 Pure Quaternion
A pure quaternion has a zero scalar term:
q
=
0
+
v
which is a vector. Therefore,
q
1
=
0
+
v
1
q
2
=
0
+
v
2
q
1
q
2
=−
v
1
·
v
2
+
v
1
×
v
2
which leads to a rather strange result for the square of a pure quaternion:
qq
=−
v
·
v
+
v
×
v
=−
v
·
v
2
=−|
v
|
a negative real number! In Hamilton's day, physicists found this result difficult to
accept, and on top of all the imaginary terms refused to adopt quaternions and em-
braced the vector analysis proposed by Gibbs
et al
.
5.6 Magnitude of a Quaternion
|
|
The
magnitude
,
norm
or
modulus
of a quaternion is written
q
and equals
q
=
s
+
x
i
+
y
j
+
z
k
s
2
x
2
y
2
z
2
.
|
q
|=
+
+
+
For example:
q
=
1
+
2
i
+
4
j
−
3
k
1
2
√
30
.
2
2
4
2
3
)
2
|
q
|=
+
+
+
(
−
=
5.7 Unit Quaternion
A unit quaternion has a magnitude equal to 1:
s
2
x
2
y
2
z
2
|
q
|=
+
+
+
=
1
.
