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.
 

1. What is the vector product in three dimensions?

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.

2. How is the vector product generalised to higher dimensions?

The vector product can be generalised to higher dimensions by using the concept of the exterior product, which involves multiplying two vectors in a higher dimensional space to create a new vector that is perpendicular to both of them.

3. What is the purpose of generalising the vector product to higher dimensions?

The generalisation of the vector product to higher dimensions allows for the application of vector algebra and geometry in spaces with more than three dimensions, which is useful in various fields such as physics, engineering, and computer graphics.

4. Can the vector product be generalised to any number of dimensions?

Yes, the vector product can be generalised to any number of dimensions. However, it is most commonly used in three dimensions, and the generalisation to higher dimensions can become increasingly complex and less intuitive.

5. Are there any limitations to generalising the vector product to higher dimensions?

One limitation of generalising the vector product to higher dimensions is that it can only be applied to vectors in Euclidean spaces, which have a fixed number of dimensions and satisfy certain geometric properties. It cannot be extended to other types of spaces, such as curved spaces in differential geometry.

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