Multidimensional cross product vector

[Olaf]

Does anyone knows how to compute cross product vector of more than 3 dimensions? It seems all the linear algebra textbooks only discuss 3D cross product vector. What are the formulas?

 

[chiro]

Does anyone knows how to compute cross product vector of more than 3 dimensions? It seems all the linear algebra textbooks only discuss 3D cross product vector. What are the formulas?

Look into the wedge product.

If you are interested in the origins read about Grassmann's framework of "Geometric Calculus" and further developments including that by Clifford.

Grassmann breaks geometric calculus into inner and outer products.

You also might want to look at linear algebra (a good graduate book), and look into books that describe tensor algebra.

 

[Olaf]

Thanks but do you know any info in the internet? I don't have those books you mentioned.

 

[Fredrik]

The cross product is only defined on $\mathbb R^3$. Wikipedia has some information about things that can be thought of as generalizations of the cross product.

http://en.wikipedia.org/wiki/Cross_product#Generalizations

 

[mathwonk]

a cross product of some vectors is supposed to be perpendiculkar to all of them and have lnegth equal to the volume they span, and be right hand oriented.

So if you want there to be only one such vector, you need to start with a product of n-1

vectors in n space. that's why you can multiply 2 of them, only in 3 space. in 4 space the cross product of three vectors makes sense.

But if you are willing to have a product which is more than one vector, maybe some kind of block, you can do it with fewer.

david bachman's book gives a nice treatment of the geometry of this subject.

 

[Olaf]

Thanks Fredrik. I found good answer from one of the external links of the wiki link that you gave: http://people.revoledu.com/kardi/tutorial/LinearAlgebra/VectorCrossProduct.html explains the code and also an interactive program.