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Homework Help: The Dot Product Expansion

  1. Mar 7, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that the dot product of vectors a and b is equal to 1/4|a+b|^2 - 1/4|a-b|^2


    2. Relevant equations
    a dot b = |a||b|cos(theta)
    a dot b = a1b1 + a2b2 + ...


    3. The attempt at a solution
    I've tried using the combination of the cosine law and those two above dot product equations and I have gotten no where. Any help would be greatly appreciated. (:
     
  2. jcsd
  3. Mar 7, 2010 #2

    Dick

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    |v|^2=v dot v for any vector v. Apply that with v=a+b and v=a-b.
     
  4. Mar 7, 2010 #3
    I've tried that, maybe there's something I'm missing but I do v=a+b and u=a-b and get them to a point where i can expand them out and I get 1/4ab as the answer.
     
  5. Mar 7, 2010 #4

    Dick

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    The ab=a dot b is right. The 1/4 isn't right. What do you get for the expansion of (a+b) dot (a+b)?
     
  6. Mar 7, 2010 #5
    (a+b) dot (a+b) = a^2 + b^2 + 2ab, yes?
     
  7. Mar 7, 2010 #6

    Dick

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    Yes. 'ab' means 'a dot b', correct? And (a-b) dot (a-b)? And the difference between the two?
     
  8. Mar 7, 2010 #7
    Haha, oh wow, I looked over that like four times.
    For some reason ab looked so wrong to me when I got it, I understand it now.

    Thanks very much (:
     
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