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Homework Help: What is cross product of two complex numbers?

  1. Dec 1, 2008 #1
    Let U=a+jb, V=c+jd.
    What is U X V? Where "X" is the cross product. Can you explain?
     
  2. jcsd
  3. Dec 1, 2008 #2
    Let U=a+jb, V=c+jd.
    What is U X V? Where "X" is the cross product. Can you explain?
     
  4. Dec 1, 2008 #3

    rock.freak667

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    UxV would give the area of the parallelogram formed by those two complex numbers.
     
  5. Dec 1, 2008 #4
    Can you please write out the steps to get the answer. I am very despirate!!!

    Thanks a million!!
     
  6. Dec 1, 2008 #5

    rock.freak667

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    It basically works out as being

    |a b|
    |c d|
     
  7. Dec 1, 2008 #6
    I believe you treat the real axis and imaginary axis as two orthogonal directions. Imagine the real component has an implied unit vector sitting out front. Think of the imiganary number j as a unit vector pointing in the imaginary direction
     
  8. Dec 1, 2008 #7
    You mean U X V = ad-bcj so Re[U X V]=ad, Im[U X V]=bc ?
     
  9. Dec 1, 2008 #8
    You mean U X V = ad-bcj so Re[U X V]=ad, Im[U X V]=bc ?
     
  10. Dec 1, 2008 #9
    I believe that's correct. Immagine if you tried to cross two real numbers. You would expect to get exactly 0 which you do.
     
  11. Dec 1, 2008 #10
    What is stil ambiguious to me is that the result of a cross product should still be a vector that is perpendicular to both U and V. What way does this vector point?
     
  12. Dec 1, 2008 #11
  13. Dec 2, 2008 #12

    Defennder

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    You mean the cross product of two vectors instead, I believe. On the Argand plane, you can treat the complex numbers as vectors. But then the area of parallelogram formed would be a real number (remember to take the absolute value as well), so you shouldn't have any imaginary part.
     
  14. Dec 2, 2008 #13
  15. Dec 2, 2008 #14
    Thanks everybody.
     
  16. Dec 2, 2008 #15
    There's still something wierd, the source I cyted ended up giving the same intuative result I had as thinking of the real axes and imaginary axes as orthogonal basis, but the real component was still imiginary. IE

    z = a+bi
    w = c+di

    z X w = yadda yadda yadda = adi - bci
     
  17. Dec 2, 2008 #16
    There are still some nice things that are comming out though. For instance, if two vectors point in the same direction in the real - imiginary plane, their cross is zero

    z = a+bi
    w = c+di


    c = 2 a d = 2 b

    z X w = adi - bci = a(2b)i - b(2a)i = 0
     
  18. Dec 2, 2008 #17
    I have been studying up the articles you gave me. The way I understand it is ZXW is pure imaginary.

    (Z'W-ZW')/2=i(ad-bc) = i[Im(Z'W)] = i[Im(ZW')]

    That is the reason it is called complex product. There is no real part from the calculation.



    The real product is Z.W=(Z'W+ZW')/2=ac+bd with no imaginary part.


    Thanks for all your help, the link you gave was a good one. I already print out all the chapters and put it in my notes.

    Good night
     
  19. Dec 2, 2008 #18
    Cross product of Complex numbers, is the book wrong or me?

    This is the exact equations derivation from the “Field and Wave Electromagnetics” by David K. Cheng which have very very few errors. But this don’t make sense. Before I conclude that this is an error, let me run this by you guys/gals.

    Re(A)=(A+A*)/2 Re(B)=(B+B*)/2
    Re(A) X Re(B) = [(A+A*)/2] X [(B+B*)/2]
    = (1/4)[(A X B* + A* X B) + (A X B + A* X B*)] (line 3).
    = (1/2) Re(A X B* + A X B). (line 4).

    I verified that A X B = (i/2)[(A* B) – (AB*)] which is pure imaginary.

    If you look at (line 3) above,
    A X B* = (i/2)[(A*B*) – (A*B)]
    A* X B = (i/2)[(AB) – (A*B*)]
    A X B = (i/2)[(A*B) – (AB*)]
    A* X B* = (i/2)[(AB*) – (A*B)]
    The sum of the four terms in (line 3) equal ZERO!!!

    As seen on (line 4), (A X B* + A X B) is pure imaginary. So Re(A X B* + A X B) = ZERO!!

    Am I missing something?
    Thanks
     
  20. Dec 2, 2008 #19
    Cross product of Complex numbers, is the book wrong or me?

    This is the exact equations derivation from the “Field and Wave Electromagnetics” by David K. Cheng which have very very few errors. But this don’t make sense. Before I conclude that this is an error, let me run this by you guys/gals.

    Re(A)=(A+A*)/2 Re(B)=(B+B*)/2
    Re(A) X Re(B) = [(A+A*)/2] X [(B+B*)/2]
    = (1/4)[(A X B* + A* X B) + (A X B + A* X B*)] (line 3).
    = (1/2) Re(A X B* + A X B). (line 4).

    I verified that A X B = (i/2)[(A* B) – (AB*)] which is pure imaginary.

    If you look at (line 3) above,
    A X B* = (i/2)[(A*B*) – (A*B)]
    A* X B = (i/2)[(AB) – (A*B*)]
    A X B = (i/2)[(A*B) – (AB*)]
    A* X B* = (i/2)[(AB*) – (A*B)]
    The sum of the four terms in (line 3) equal ZERO!!!

    As seen on (line 4), (A X B* + A X B) is pure imaginary. So Re(A X B* + A X B) = ZERO!!

    Am I missing something?
    Thanks
     
  21. Dec 2, 2008 #20

    D H

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    Re: Cross product of Complex numbers, is the book wrong or me?

    Kind of roundabout way to get to that result, but this is indeed a tautology for any two complex numbers A and B.

    A*B-AB* is pure real because AB* = (A*B)*. However, your equality is not valid. Show your derivation.
     
  22. Dec 2, 2008 #21
    Re: Cross product of Complex numbers, is the book wrong or me?

    There's a lot going on there in your post, so I didn't look at it much. Here's a link that shows various forms for the cross product of two complex numbers--see bottom portion of page 6.

    http://books.google.com/books?id=9zKl4lXEXlsC&pg=PA6

    You can compare it with your book.
     
  23. Dec 2, 2008 #22

    D H

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    Re: Cross product of Complex numbers, is the book wrong or me?

    That stuff in Schaum's Outline is non-standard nomenclature. AxB typically denotes the complex product, not the "cross product". I don't have Field and Wave Electromagnetics, but I would suggest you first check that they aren't discussing the complex product there.

    BTW, even if they are using the cross product, the result is still valid because the cross product of two pure real numbers (or two pure imaginary numbers for that matter) is identically zero.
     
  24. Dec 2, 2008 #23
    Re: Cross product of Complex numbers, is the book wrong or me?

    I don't think so, this is the equation:

    Let A=a+bi, B=c+di

    A X B= (a-bi)(c+di)-(a+bi)(c-di)=[(ac+bd)+i(ad-bc)] - [(ac+bd)+i(bc-ad)]
    =i[2(ad-bc)] this is pure imaginary
     
  25. Dec 2, 2008 #24
    Re: Cross product of Complex numbers, is the book wrong or me?

    I have the book, I verify all my formulas already. In fact I got this link from a person here which is much better than Scharms:

    http://www.cut-the-knot.org/arithmetic/algebra/RealComplexProducts.shtml

    THanks
     
  26. Dec 2, 2008 #25

    D H

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    Re: Cross product of Complex numbers, is the book wrong or me?

    That was a stupid typo, sorry. A*B-AB* is obviously pure imaginary. Sorry about that.

    Have you verified what Cheng means by the symbol 'x'? I've look through a couple of my complex analysis texts and a couple of my EM texts. None of them use the concept of a complex cross product.

    Regardless, the relationship is true for both the complex product and the cross product.

    Edit
    I think you should double-check the nomenclature in Cheng. Why in the world would you take the real part of an expression that is by definition pure imaginary (the cross product)?
     
    Last edited: Dec 2, 2008
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