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Law of Sines Proof w/ Vector Calculus

  1. Feb 18, 2007 #1
    Hey guys, it's my first post here so please don't chew my head off if I do something forbidden, hahah.

    1. The problem statement, all variables and given/known data

    Prove the Law of Sines using Vector Methods.

    2. Relevant equations

    sin(A)/a = sin(B)/b = sin(C)/c

    3. The attempt at a solution

    Since axb=sin(C), I decided to try getting the cross product and then trying to match it to the equation. I gave each of the points coordinates and I decided to come here for help when the problem became gigantic and hardly manageable. I've spent a couple of hours on this and I'm not getting anywhere.

    If you've got any suggestions for a simpler solution or even just a nudge in the right direction, I'd be very grateful.
  2. jcsd
  3. Feb 18, 2007 #2


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    Here's a hint:
    Start by drawing a triangle with vectors, a, b, c.
    Can you apply the cross-product with any pair of these vectors?
  4. Feb 19, 2007 #3
    Yay, got it! Thanks!

    I said
    axb = area of parallelogram
    bxc = area of parallelogram
    cxa = area of parallelogram

    so axb = bxc = cxa
    then absin(C) = bcsin(A) = casin(B)

    then crossed stuff out and got the law of sines.
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