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Proof using vector geometry, I need some direction please.

  1. Aug 30, 2011 #1
    1. The problem statement, all variables and given/known data

    Ok, this is actually an assignment from my calc III class but it involves vectors so I was hoping I could get some help here.

    The problem is There are two vectors A and B that make a parallelogram. I have to prove that when two line segments are drawn from a vertice from the vectors that bisect the vectors on the other side of the parallelogram the line segments will trisect the diagonal between the vectors. One of the tri-sections is labeled vector C. I am attaching a picture that I drew in paint which I think illustrates the problem. If it doesn't then It is how I understood it and that might be why I am having trouble to begin with. I have scribbled down a few things that I know are true, but I am really having a problem getting started.

    2. Relevant equations

    Relevant info is all rules pertaining to vectors and geometry and trig. I'm sure I'm just overlooking something simple.

    3. The attempt at a solution

    I have tried starting with the assumption that A+B= 3C

    1/2A+B= one of the line segments

    A+ 1/2B= the other line segment

    these are things that I think maybe I need to assume to be true but I honestly have no idea where to start. Could someone please point me in the right direction? The diagram is really bad btw and not to scale, (like I said, I drew it in paint).
     

    Attached Files:

  2. jcsd
  3. Aug 31, 2011 #2

    PeterO

    User Avatar
    Homework Helper

    Interesting that my first thought was that one of the two extra lines you added was not as described, but no matter because:

    My descriptions may be a little unconventional here but I hop you understand.

    You drew a line from A/2 to B - I expected that
    I expected the other line to be from A to B/2.

    However the line you drew from the "mid-point of the top" to B could represent exactly the same vector, just translated b/2 to the right.

    by then taking that C divided those two new lines in the ratio m:n, I could show than m:n was in the ration 1:2, so indeed C was a third of the full diagonal.

    A starting point is to give vector notation to the new lines you drew.

    You could start by describing the line you drew from B through C to half way along A as -B +A/2 if you consider one direction, or -A/2 + B if you take the other direction.

    Consider that C divides that new vector in the ratio m:n [little bit to big bit] so the short part is m/(m+n) of the full vector, while the other bit is n/(m+n) of the full vector. Substitute and have fun.
     
    Last edited: Aug 31, 2011
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