How to genarelize the vector product in three dimensions to higher dimensions?

In summary, the vector product, also known as the cross product, is a mathematical operation between two vectors in three-dimensional space that results in a third vector that is perpendicular to the original two vectors. It can be generalised to higher dimensions using the concept of the exterior product, allowing for the application of vector algebra and geometry in spaces with more than three dimensions. However, the generalisation to higher dimensions can become increasingly complex and less intuitive. Additionally, it can only be applied to vectors in Euclidean spaces and cannot be extended to other types of spaces.
  • #1
wdlang
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it seems that the vector product between vectors in three dimensions is peculiar property of the three dimensional space
 
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  • #2
  • #3
wdlang said:
it seems that the vector product between vectors in three dimensions is peculiar property of the three dimensional space

That is true.

If one has a metric you can talk about volumes and hypervolumes and come up with an a hypercube orthogonal to any given hypercube of complementary dimension. If you have an orientation then an oriented hypercube will have an oriented complement.

This all can be done with wedge product one you have an orthonormal basis.
 

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