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

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The vector product is a unique characteristic of three-dimensional space, with its properties not directly applicable in higher dimensions. In higher dimensions, one can utilize metrics to discuss volumes and hypervolumes, leading to the concept of hypercubes orthogonal to complementary dimensions. An oriented hypercube can also have an oriented complement when orientation is considered. The wedge product becomes useful in these higher-dimensional contexts, particularly when an orthonormal basis is established. Understanding these concepts is essential for generalizing the vector product beyond three dimensions.
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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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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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