Cross product in higher dimensions?

[waht]

Although, the dot product works in infitine dimensions, it is not the case for the cross product. Anybody know in what dimensions the cross product can be defined?

 

[dextercioby]

The cross product is a particular case of a wedge product.(In 3D,it is the Hodge dual of the wedge product between 2 1-forms).Of course,no one can put a limit on the dimension of the manifold,but customarily,this manifold is finite dimansional.

Daniel.

 

[matt grime]

AKA exterior product or occasionally outer product in some algebraic texts. Look up exterior algebra.

 

[mathwonk]

the cross product v x w can be defined by determinants and dot products, as the unique vector u such that for every vector z, we have

u.z = det[v,w,z].

then this generalizes to all finite dimensions n as follows:

the cross product of the n-1 vectors v1,...,vn-1, is the unique vector u such that for all vectors z we have

z.u = det[v1,...,vn-1,z].

since this is a product not of two vectors but of n-1 of them, it is not considered a "product" by everyone.

it does occur however as the "triple product" in vector analysis (v x w).z.


for this definition, see spivak, calculus on manifolds, page 84.



the relation with the exterior products, is due to the fact that exterior products are an intrinsic way to write determinants.

i.e. if we write wedge ^ for exterior product, then we can multiply v1^...^vn-1 and get an object whose wedge product with any vector is an element of the n th wedge of R^n, hence naturally a number, since it equals a unique scxalar times the wedge of the stabndard unit vectors e1^...^en.

thus wedging with v1^...^vn-1 is the same as dotting with something which we could call the cross product of the v1,..,vn-1.

I think this is what is meant by saying the cross product is the "Hodge dual" of the wedge v1^...^vn-1.

 

[HallsofIvy]

A standard way of defining dot product in other dimensions than 3 is to use the "alternating tensor": A_ijkl... (where the number of indices is the same as the dimension of the space) is defined as: 0 if any of the indices repeat. If not then ijkl... is a permutation of 123...n. A_ijkl...= 1 if that permutation is even, -1 if it is odd.

In 3 dimensions Auv= A_ijku_jv_k ("contracting" on j and k) is the cross product of u and v.

In other dimensions, the cross product of vector u₁, u₂, ..., u_n-1 (1 less than the number of dimensions) is Au₁u₂...u_n-1.