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Cross product of difference and sum of two vectors

  1. Aug 26, 2011 #1
    1. The problem statement, all variables and given/known data
    show that:
    ( a - b ) x (a + b ) = 2a x b

    and wat is its geometric interpretation ??
    I'm not sure what's wrong, but i somehow got the value as 2a and not wat was required... PLease help.




    2. Relevant equations
    Since this is a proof, the answer i've arrived at is wrong. How do I arrive at the solution ?


    3. The attempt at a solution
    First of all, i took the cross product and tried proving the statement, but couls arrive at it whatsoever. I ended up with a 2a actually.
    As for the second part, with the parallelogram idea, i really could not figure out how to find 2a x b. After analysis, I arrived at 2a , but am stuck there . please help.
     
  2. jcsd
  3. Aug 26, 2011 #2
    Can you show your work? I get the answer required in two lines of algebra.
     
  4. Aug 26, 2011 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Geometrically, a+ b and a- b are the diagonals of a parallelogram having a and b as sides. The geometric content of your equation is "If a and b are sides of a parallelgram, P, then the parallelogram having its diagonals as sides has area twice the area of P".
     
  5. Aug 26, 2011 #4
    thanks hallsofivy..., now it appears more clear ... i messed up my vector diagrams so badly i didn't see the actual thing .. for the first part, i got 2 (a x b). is it equal to
    2a x b ??
     
  6. Aug 26, 2011 #5
    No, 2(a x b) is not same as 2a x b. :smile:

    And the question you have posted should be:-
    show that : (a-b)x(a+b)=2(a x b)
     
  7. Aug 26, 2011 #6
    should be then .. but the textbook says 2a x b. or maybe i read it wrong !!
     
  8. Aug 26, 2011 #7

    I like Serena

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    Homework Helper

    Sorry Pranav-Arora, but 2(a x b) = (2a) x b = 2a x b.
    The cross product is a linear operator.

    Note that a x a = 0 and that a x b = - b x a.
     
  9. Aug 26, 2011 #8
    Sorry, You're right. :smile:
     
  10. Aug 26, 2011 #9
    thanks people :)
     
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