Cross product of complex vectors

In summary, the conversation is about a vector product, specifically the second equation derived from the first one. The author's derivation seems to be missing a -2i factor in the second term of the second equation. The question is how to verify that the cross product of a complex vector and its conjugate is equal to the cross product of the real part of the complex vector and its imaginary part. The author's simulation is based on the second equation and it seems to make sense, but the first equation does not. The conversation is from a book by Morgan and Green titled "AC electrokinetics: Colloids and nanoparticles".
  • #1
s_guo82
2
0
Would you pls help me with the following vector product? I got no idea how the author derived the second equation from the first one. My derivation result is always including the imaginary unit i for the second term in the second equation on the right hand side. Specifically, how to verify that the cross product of the complex vector and its conjugate is equal to the cross product of the real part of the complex vector and the its imaginary part? ~ denotes a complex variable.
equ mor and gre.png


Thank you in advance
 
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  • #2
Looks like a -2i factor is missing in the second term of the second equation. You get this as well?
 
  • #3
Yes, I got the same thing. I doubt there should be a "i" for the second term on the right hand side of the second equation. However, all the author's simulation is based on the second equation. I also tried to expand the first equation and simulated it, but it did not give a result that makes sense. However, with the second equation, the simulation result seems to make sense. I asked this question because I am neither a maths student nor physics and I am not confident that the renowned author can make mistakes for such an important expression. He used this equation all the way in his book.

This is from the book "AC electrokinetics: Colloids and nanoparticles" by Morgan and Green.

Any other thoughts, guys?
 

1. What is the cross product of complex vectors?

The cross product of two complex vectors is a vector that is perpendicular to both input vectors in 3D space. It is represented by a complex number and can be calculated using the determinant of a 2x2 matrix.

2. How is the cross product of complex vectors calculated?

The cross product can be calculated by taking the determinant of a 2x2 matrix with the input vectors as its columns. The result will be a complex number representing the cross product vector.

3. What are the properties of the cross product of complex vectors?

The cross product of complex vectors follows the same properties as the cross product of real vectors. It is anti-commutative, distributive, and satisfies the right-hand rule for determining its direction.

4. What are the uses of the cross product of complex vectors?

The cross product of complex vectors is used in many fields of science and engineering, such as electromagnetism, fluid dynamics, and computer graphics. It can be used to calculate torque, magnetic fields, and 3D rotations.

5. Can the cross product of complex vectors be visualized?

Yes, the cross product of complex vectors can be visualized in 3D space as a vector that is perpendicular to both input vectors. Its magnitude can be seen as the area of a parallelogram formed by the two input vectors, and its direction can be determined using the right-hand rule.

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