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Math Help - Three vectors a, b and c such that a+b+c=0

  1. #1
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    Three vectors a, b and c such that a+b+c=0

    Hi, I'm having trouble with the second part of this question and could really use a nudge in the right direction...

    Let a, b and c be three vectors such that a+b+c=0. Show that a x b = b x c = c x a and explain what this means geometrically.

    I've proved that a x b = b x c = c x a by means of expanding the two parts of the equation and substituting values in etc, but I'm stuck on the second bit. I know that a+b+c =0 means that a, b and c form a triangle... but beyond that I'm not sure what else to say. How do I relate this back to a x b = b x c = c x a?

    Your help would be greatly appreciated. Cheers!
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  2. #2
    Senior Member BAdhi's Avatar
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    you've understood that a, b, c form a triangle, so this means a, b, c vectors are on the same plane.

    now, you know when we take a cross product of two non linear vectors, it will geometrically give the area of the parallelogram they make. also the half of that value would give the area of the triangle they make. so, axb, bxc and cxa refers to the same triangle so the moduls value of all three should be equal and since a,b,c are on the same plane, the direction of those cross products also should be the same which make those cross products equal
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  3. #3
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    Cheers, BAdhi, that clears things up alot.
    I really appreciate your quick reply, thank you! =)
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