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Properties of Dot Product

  1. Feb 17, 2013 #1
    1. The problem statement, all variables and given/known data 3. The attempt at a solution


    I am working a physics problem and want to make sure I'm not making a mistake in the math. Here is my math inquiry:

    Say you have (a*b)(c*d) where * indicates the dot product, and a,b,c, and d are all vectors. Can you say that (a*b)(c*d) = (a*c)(b*d) since the dot product is commutative and gives you a scalar? Thanks!
     
  2. jcsd
  3. Feb 17, 2013 #2

    Dick

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    No. You can't. Neither of those properties says you can swap vectors between two different dot products.
     
  4. Feb 17, 2013 #3
    If you're ever unsure of these things, always try to find a counterexample. Take a = [1,0], b = [1,0], c = [1,1] and d = [1,2]. Then,

    (a*b)(c*d) = (1)(1 + 2) = 3

    but

    (a*c)(b*d) = (1)(1) = 1
     
  5. Feb 17, 2013 #4
    Sorry, accidentally double posted.
     
  6. Feb 17, 2013 #5
    Darn- it made the problem so easy, but I guess that was a sign that I was probably doing something that I should not be doing. Thank you for explaining why I could not shuffle the vectors around. :)

    I did try an example, however it worked out. But I guess it wasn't that good of an example since it was a fluke that it worked out. Thanks
     
  7. Feb 17, 2013 #6

    Ray Vickson

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    No, as others have already pointed out.

    You can see this another way: a*b = |a||b| cos(a,b) and c*d = |c||d|cos(c,d), where |a|, |b| are the magnitudes of a and b and (a,b) is the angle between a and b, and similarly for c and d. In general, we do not have cos(a,c).cos(b,d) equal to cos(a,b).cos(c,d), so the two expressions are generally different.
     
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