Possible Generalization of Cross Product?

In summary, there is a formula for finding a vector perpendicular to two given vectors in C^3 using the cross product defined by the wedge product and Hodge star. However, it is essentially the same as the formula for the cross product in R^3, with the only difference being that the components of the vectors are complex numbers. The formula can be written in a matrix form as a determinant, making it easier to remember, and follows a distinct pattern of "231".
  • #1
SeReNiTy
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In R^3 it is easy to compute the cross product, and i know how to compute it in higher dimensions using wedge product and the hodge star, which shows that it only exists in 3n dimensions.

My question is given two vectors in C^3 (complex), is there a neat way to find one perperdicular to both? Ie - a generalization of the original cross product?
 
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  • #2
I don't think that's possible. If you define two vectors that are perpindicular to a third by setting the dot product to zero. Then subtitute and solve the three systems of equations... then you get the cartesian cross product equation. That is the proof that the cross product equation is the general situation of having a vector perpindicular to two others. Therefore, I'll assume the equation is the simplest method.
 
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  • #3
And the result turns out to be essentially the same as with real vectors. (I found the formula a different way -- simply starting with the ordinary cross product, and seeing what needs to be changed)
 
  • #4
So what is the formula guys? I know there must exist one since C^3 is just a six dimensional real space, and i know cross products are defined in 3n space according to the wedge product definition.
 
  • #5
Don't worry guys i solved it using the wedge product and Hodge star, it turns out that aXb is exactly the same as the normal definition in R^3 except you conjugate both (a) and (b) before you do the cross product.

where a = (a1,a2,a3) and b = (b1,b2,b3) and the components themselves are elements of C, ie a1 = x+yi
 
  • #6
Oh, you didn't even know the formula?

[tex] A X B = [a_{2}b_{3} - a_{3}b_{2}, a_{3}b_{1} - a_{1}b_{3}, a_{1}b_{2} - a_{2}b_{1}] [/tex]

It may look complicated, but there is a distinct pattern that's very simple to remember. "231".
 
  • #7
Sane said:
Oh, you didn't even know the formula?

[tex] A X B = [a_{2}b_{3} - a_{3}b_{2}, a_{3}b_{1} - a_{1}b_{3}, a_{1}b_{2} - a_{2}b_{1}] [/tex]

It may look complicated, but there is a distinct pattern that's very simple to remember. "231".

Heh, of course i knew that, but i just wanted to see if there was a complex version for C^3. Btw, its easier to remember by writing in matrix form and taking it as a determinant.
 
  • #8
Don't worry guys i solved it using the wedge product and Hodge star
Yay! I figured you could do it.
 

1. What is a cross product?

A cross product is a mathematical operation that combines two vectors to create a new vector that is perpendicular to both of the original vectors. It is commonly used in physics and engineering to calculate the direction and magnitude of a force or torque.

2. How is the cross product calculated?

The cross product of two vectors, A and B, is calculated by taking the determinant of a 3x3 matrix. The resulting vector is written as A x B and is perpendicular to both A and B. The magnitude of the cross product is equal to the area of the parallelogram formed by A and B.

3. What is the purpose of generalizing the cross product?

The generalization of the cross product aims to extend its use beyond just three-dimensional space. By introducing new mathematical concepts, such as quaternions and geometric algebras, the cross product can be applied in higher dimensions and different mathematical contexts.

4. Are there limitations to the cross product?

Yes, the cross product is only defined in three-dimensional space and cannot be extended to higher dimensions. It also has certain restrictions, such as the fact that it only applies to vectors and not to other types of mathematical objects.

5. How is the cross product used in real-world applications?

The cross product has various applications in physics, engineering, and computer graphics. It is used to calculate torque in mechanical systems, to determine the direction of magnetic fields, and to create 3D computer graphics. It is also used in navigation systems, robotics, and many other areas of science and technology.

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