Vector space for cross products?

pivoxa15
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What vector space are cross products done in?
 
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If you're talking about the standard definition, it's only defined for vectors in three dimensions, so anything that has three dimensions you could do the cross product in. Why?

It occurs to me that I think there's a more general form for other dimensions, but I don't know it
 
A more general definition is this: Let \epsilon_{ijk...m} be the "alternating tensor" in m dimensions: +1 if ijk...m is an even permutation of 123...n, -1 if an odd permutation, 0 otherwise. Then we can define the "cross product" of n-1 vectors v_1, v_2, ..., v_{n-1} to be the vector v= \Sigma \epsilon_{ij...m}v_{1i}v_{2j}...v_{n-1,m} where the sum is take over repeated indices. If n= 3 then that gives the cross product on R2.
 
given n-1 vectors in n dimensions, let them act on another vector w by taking the detrminant of the mtrix the n vectors form togetehjr. that gives alinear \map of w, which is thus dotting with aunique vector caled the cross product of the first n-1 vectors.
 

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