Graphics Reference
In-Depth Information
For example:
q
1
=
0
.
6
+
2
i
+
4
j
−
3
k
q
2
=
+
+
+
0
.
2
3
i
5
j
7
k
q
1
+
q
2
=
+
+
+
0
.
8
5
i
9
j
4
k
q
1
−
q
2
=
0
.
4
−
i
−
j
−
10
k
.
5.4 Multiplying Quaternions
When multiplying quaternions we must employ the following rules:
i
2
j
2
k
2
=−
1
,
=−
1
,
=−
1
,
ijk
=−
1
=
=
=
ij
k
,
jk
i
,
ki
j
=−
=−
=−
j
.
Note that quaternion addition is commutative, however, the rules make quaternion
products non-commutative. For example:
ji
k
,
kj
i
,
ik
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
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
=
s
1
s
2
−
(x
1
x
2
+
y
1
y
2
+
z
1
z
2
)
+
s
1
(x
2
i
+
y
2
j
+
z
2
k
)
+
s
2
(x
1
i
+
y
1
j
+
z
1
k
)
x
2
y
1
)
k
which can be rewritten using the dot and cross product notation as
q
1
q
2
=
+
(y
1
z
2
−
y
2
z
1
)
i
+
(z
1
x
2
−
z
2
x
1
)
j
+
(x
1
y
2
−
s
1
s
2
−
v
1
·
v
2
+
s
1
v
2
+
s
2
v
1
+
v
1
×
v
2
where
s
1
s
2
−
v
1
·
v
2
is a scalar, and
s
1
v
2
+
s
2
v
1
+
v
1
×
v
2
is a vector.
For example:
q
1
=
1
+
2
i
+
3
j
+
4
k
q
2
=
2
−
i
+
5
j
−
2
k
q
1
q
2
=
(
1
+
2
i
+
3
j
+
4
k
)(
2
−
i
+
5
j
−
2
k
)
q
1
q
2
=
1
2
)
×
2
−
2
×
(
−
1
)
+
3
×
5
+
4
×
(
−
+
1
(
−
i
+
5
j
−
2
k
)
+
2
(
2
i
+
3
j
+
4
k
)
+
3
5
i
−
2
1
)
j
+
2
1
)
k
×
(
−
2
)
−
4
×
×
(
−
2
)
−
4
×
(
−
×
5
−
3
×
(
−
=−
3
+
3
i
+
11
j
+
6
k
−
26
i
+
13
k
=−
3
−
23
i
+
11
j
+
19
k
