This page discusses the usual cross product axb of two vectors a,b in R^3,
and its generalizations in higher dimensions R^n, n > 3.
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A product axb satisfies the usual rules ax(b+c) = axb+axc, (a+b)xc = axc+bxc,
and ax(kb) = (ka)xb = k(axb), where a and b are vectors in a real linear space
V and k is a real number in R. Consider a Euclidean space V = R^n.
Definition: A product axb of two vectors a,b in V is a cross product, if
1. axb is a vector orthogonal to both a and b, and
2. the length of axb is equal to the area of the parallelogram of a and b.
Theorem: A cross product of two vectors exists only in R^3 and R^7.
In R^3, there are two choices for the orientation of a, b, axb.
Multiplication rules of the cross product in R^7, with basis (e1,e2,...,e7),
are determined by antisymmetry eixej = -ejxei and the rules
e1xe2 = e4, e2xe4 = e1, e4xe1 = e2,
e2xe3 = e5, e3xe5 = e2, e5xe2 = e3,
. . .
. . .
e6xe7 = e2, e7xe2 = e6, e2xe6 = e7,
e7xe1 = e3, e1xe3 = e7, e3xe7 = e1.
Both in R^3 and in R^7, there is a connection between the exterior product
a^b of two vectors a,b and the cross product axb, namely
axb = -(a^b)_|v,
where v is a 3-vector and "_|" denotes the contraction. In 3D, v = e1e2e3,
and in 7D, v = e124+e235+e346+e457+e561+e672+e713.
Thus, dim(V) = dim(V^V), or n = n(n-1)/2, is not a necessary condition for
the existence of a cross product of two vectors in V, dim(V) = n.
In R^3, there are only two choices for axb, depending on the orientation,
chosen by a unit volume element v of R^3. In R^7, there is an infinity
of choices for axb, depending on the choice of a 3-vector v of R^7.
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The cross product in 3D can be defined in terms of coordinates as follows:
Definition: The cross product c = axb of two vectors a and b is given by
(c1,c2,c3) = (a2b3-a3b2,a3b1-a1b3,a1b2-a2b1),
where ck is the k-th coordinate of c. This can be condensed as
ck = sum_{i,j} e(i,j,k) ai bj,
where e(i,j,k) is 0 if any of the indices i,j,k are the same, 1 if (i,j,k)
is an even permutation of (1,2,3) and -1 if (i,j,k) is an odd permutation.
This definition can be generalized to higher dimensions, to a product of
n-1 vectors in R^n, n > 1.
Definition: The cross product c = a_1x...xa_{n-1} of vectors a_1,...,a_{n-1}
in dimension n is a vector with coordinates
c{i_n} = sum_{i_1,...,i_{n-1}} e(i_1,...,i_n) a_1{i_1} ... a_{n-1}{i_{n-1}}
where e(i_1,...,i_n) is a multi-indexed quantity that is 0 if any two indices
are the same, 1 if (i_1,...,i_n) is an even permutation of (1,...,n), and -1 if
(i_1,...,i_n) is an odd permutation. The term a_k{i_j} denotes the {i_j}-th
coordinate of the k-th vector.
Example in 2D: (c1,c2) = (e(1,1)a1+e(2,1)a2,e(1,2)a1+e(2,2)a2) = (-a2,a1),
cj = sum_i e(i,j) ai.
This definition is non-intrinsic, while it depends on a basis and coordinates.
That shortcoming can be circumvented as follows:
Definition: The cross product of n-1 vectors a_1,a_2,...,a_{n-1} in nD
is the vector
a_1xa_2x...xa_{n-1} = -(a_1^a_2^...^a_{n-1})_|e12...n.
where e12...n = e1e2...en is the unit volume element in nD.
This intrinsic definition still has the defect that it does not include, or
coincide with, the cross product two vectors in 7D.
The following is a definition of a generalized cross product, including
both the cross product of two vectors in R^n, n = 3 or 7, and the cross
product of n-1 vectors in R^n:
Definition (generalized): The cross product of k vectors is a vector
perpendicular to the k vectors and of length equal to the k-volume
of the parallelepiped of the k vectors.
Theorem: The cross product of k vectors in R^n exists only for the following
pairs (n,k): (3,2), (7,2), (8,3), (n,n-1) and when n is even also for (n,1).
The proof of existence of the cross product of two vectors in 7D can be based
on octonions, O, in the same way as the 3D cross product can be based on
quaternions, H. Let D be H = R+R^3 or O = R+R^7. Then, the cross product
axb of two vectors a and b in Ve(D) is
axb = Ve(ab),
where ab is the quaternion/octonion product of a and b, and Ve(ab)
is the vector part of ab.
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