Graphics Reference
In-Depth Information
cross product is a “product” that behaves very much like the product in the case of
real numbers except that it is not commutative. The two operations of vector addi-
tion and the cross product make
R
3
into a (noncommutative)
ring
. Is there a similar
product in other dimensions? Unfortunately not, but the cross product does arise from
a general construction that applies to all dimensions and that is worth looking at
because it will give us additional insight into the cross product.
Let
v
1
,
v
2
,...,
v
n-1
Œ
R
n
. Define a map T :
R
n
1.10.1. Theorem.
Æ
R
by
v
Ê
ˆ
1
Á
Á
Á
˜
˜
˜
M
()
=
T
w
det
.
v
n
-
1
Ë
¯
w
Then there is a unique
u
Œ
R
n
such that T(
w
) =
u
•
w
for all
w
.
Proof.
This theorem is an immediate corollary to Theorem 1.8.2 because properties
of the determinant function show that T is a linear functional.
Definition.
Using the notation of Theorem 1.10.1, the vector
u
is called the (
gener-
alized
)
cross product
of the vectors
v
1
,
v
2
,...
v
n-1
and is denoted by
v
1
¥
v
2
¥ ···¥
v
n-1
.
1.10.2. Proposition.
The generalized cross product satisfies the following basic
properties:
(1) It is commutative up to sign, that is,
()
¥
v
¥
v
¥◊◊◊¥
v
=
sign
s
v
v
¥◊◊◊¥
v
()
()
(
)
s
1
s
2
s
n
-
1
1
2
n
-
1
for all permutations s of {1, 2,..., n - 1}.
(2) It is a multi-linear map, that is,
v
¥◊◊◊¥
a
v
¥◊◊◊¥
v
=
a
(
v
¥◊◊◊¥
v
¥◊◊◊¥
v
)
1
i
n
-
1
1
i
n
-
1
(
)
¥◊◊◊¥
¢
(
)
+
(
)
v
¥◊◊◊¥
v
+
v
v
=
v
¥◊◊◊¥
v
¥◊◊◊¥
v
v
¥◊◊◊¥
v
¢ ¥◊◊◊¥
v
1
i
i
n
-
1
1
i
n
-
1
1
i
n
-
1
(3) (
v
1
¥
v
2
¥ ···¥
v
n-1
)•
v
i
= 0 , for all i.
(4) If the vectors
v
i
are linearly independent, then the ordered basis
(
)
vv
,
,
◊◊◊
v
,
v v
¥
¥◊◊◊¥
v
12
n
-
11
2
n
-
1
induces the standard orientation on
R
n
.
Proof.
Facts (1) and (2) are immediate from the definition using properties of the
determinant. Fact (3) follows from the observation that the determinant of a matrix
with two equal rows is zero, so that each
v
i
lies in the kernel of T in Theorem 1.10.1.
