Graphics Reference
In-Depth Information
and
2
(
)
.
2
2
2
v v v
i
++
j
k
=-++
vvv
(20.2b)
1
2
3
By equation (20.2a) the norm of a quaternion is just the standard length when it is
thought of as a vector in
R
4
, and
20.2.4
Proposition.
Let
a
,
b
Œ
H
.
(1) |
a
| = | .
(2) |
ab
| = |
a
| |
b
|.
(3) If
a
π
0
, then
a
-1
exists and
a
-1
a
= |
a
|
-2
. Furthermore, |
a
-1
| = |
a
|
-1
.
a
Proof.
Exercise 20.2.4.
Definition.
A quaternion
a
is said to be a
unit quaternion
if |
a
| = 1.
The set of unit quaternions is just the unit sphere
S
3
in
R
4
. It
20.2.5
Proposition.
is a subgroup of
H
.
Proof.
Exercise 20.2.5.
Definition.
Let
a
and
b
be pure quaternions. The
vector product
a
¥
b
of
a
and
b
is
defined by
¥=
()
ab
pu
ab
.
Since we have identified pure quaternions with
R
3
, we can think of the function
¥ as defining a product on
R
3
. The next proposition justifies the notation since
a
¥
b
agrees with the usual cross product. One way to look at this is to treat the new defi-
nition basically as an alternate algebraic definition for the cross product.
20.2.6
Proposition.
Let
a
and
b
be pure quaternions.
(1) The map ¥:
R
3
¥
R
3
Æ
R
3
that sends (
a
,
b
) to
a
¥
b
is bilinear.
(2)
ab
=-
a
•
b
+
a
¥
b
(3)
a
¥
b
=-(
b
¥
a
)
(4)
a
¥
a
=
a
•(
a
¥
b)
=
b
•(
a
¥
b
) =
0
(5) In terms of our identification of pure quaternions with
R
3
,
a
¥
b
is just the
ordinary cross product of
R
3
.
Proof.
This is again a straightforward computation that is left as Exercise 20.2.6.
Proposition 20.2.6 (2) leads to an alternate definition in closed form of the quater-
nionic product where one uses only basic vector operations:
20.2.7
Corollary.
If
a
= r +
v
and
b
= s +
w
are two quaternions, where r,s Œ
R
and
v
,
w
Œ
R
3
, then
ab
=-∑+ ++¥
rs
v w
r
w
s
v
(
v
w
)
.
