No Cross Product in higher dimensions?

In summary, the conversation discusses the existence of a direct analogue of the binary cross product in general dimensions and the Wedge Product as an alternative. The concept of perpendicular vectors in higher dimensions is also examined, with the conclusion that it is only applicable in three dimensions. The article mentioned provides further information on vector products in spaces with more than 3 dimensions.
  • #1
MathewsMD
433
7
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 RN?

Any explanation is greatly appreciated!
 
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  • #2
MathewsMD said:
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 RN?
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?
 
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  • #3
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.
 
  • #4
MathewsMD said:
Is it really unknown on how to find a perpendicular vector to any vector in RN?

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.
 
  • #6

1. What is the cross product in higher dimensions?

The cross product is a mathematical operation that takes two vectors in three-dimensional space and produces a third vector that is perpendicular to both of the original vectors.

2. Why is there no cross product in higher dimensions?

The cross product is only defined in three dimensions because it relies on the concept of a perpendicular vector, which does not exist in higher dimensions. In higher dimensions, there are an infinite number of possible vectors that could be considered perpendicular, making it impossible to define a unique cross product.

3. How is the cross product related to the dot product?

The cross product and the dot product are two different operations between vectors. The dot product results in a scalar value, while the cross product results in a vector. They are related in that the dot product of two perpendicular vectors is always zero, and the cross product of two parallel vectors is always zero.

4. Are there alternative methods for calculating the cross product in higher dimensions?

Yes, there are alternative methods for calculating the cross product in higher dimensions, such as using the exterior product or the wedge product. These methods involve using multivectors and can produce similar results to the cross product in three dimensions.

5. How does the absence of the cross product affect higher-dimensional mathematics?

The absence of the cross product in higher dimensions does not significantly affect mathematics, as there are alternative methods for calculating similar results. However, it does limit the use of certain geometric concepts and operations that rely on the cross product, such as the calculation of the area of a parallelogram or the torque of a force in physics.

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