No Cross Product in higher dimensions?

[MathewsMD]

Is there an intuitive reason or proof demonstrating that in general dimensions, there is no direct analogue of the binary cross product that yields specifically a vector?

I came across Wedge Product as the only alternative, but am just learning linear algebra and don't quite comprehend yet why this isn't exactly considered a regular vector.

Is it really unknown on how to find a perpendicular vector to any vector in R^N?

Any explanation is greatly appreciated!

 

[Simon Bridge]

Is there an intuitive reason or proof demonstrating that in general dimensions, there is no direct analogue of the binary cross product that yields specifically a vector?

Check the definition of the cross product.

Note: it is possible to describe a plane surface in n>3 dimensions, and thus to find two vectors in the surface that are not colinear, and thus find the normal vector to the surface via an operation on the two vectors in the surface. Would this count as an n-D analogue for a cross product?

Well...
https://www.physicsforums.com/showthread.php?t=526403

I came across Wedge Product as the only alternative, but am just learning linear algebra and don't quite comprehend yet why this isn't exactly considered a regular vector.

The wedge product is an operation on two vectors ... to see why the result is not a vector, just apply your new-found knowledge of what a vector is and see.

Is it really unknown on how to find a perpendicular vector to any vector in R^N?

Check the definition of "perpendicular". Does a 4D vector describe an object for which something can be "perpendicular" ... how does the concept make sense in more than 3D?

 

[WWGD]

I think it has to see with some special properties of the Hodge dual in dimension 3. http://en.wikipedia.org/wiki/Hodge_dual

Read the paragraph right above 'Extensions' in that page.

 

[The_Duck]

Is it really unknown on how to find a perpendicular vector to any vector in R^N?

In three dimensions you can pick two vectors A and B and ask for a vector C that is perpendicular to both A and B. This vector C is unique up to a sign. (Except in the special case that A and B are collinear).

This only works in three dimensions. In two or fewer dimensions, there are no vectors perpendicular to both A and B. In four or more dimensions, there are an infinite number of vectors perpendicular to both A and B.

 

[SteamKing]

This article has a lengthy discussion about vector products in spaces with more than 3 dimensions:

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

 

[pwsnafu]

Cross product also exists in seven dimensions. It is not unique though.