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Math Help - Betweeness of points

  1. #1
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    Betweeness of points

    Hi, I am just working on random problems to study for my midterm, but then I came across this one I don't know how to do. Plz help me. Thankkk youuu sooo much.


    1) Suppose that A, B, C are collinear points in R^3 whose coordinates are given by (a1, a2, a3), (b1, b2, b3), and (c1, c2, c3) respectively. Prove that A*B*C holds if we have a1 < b1 < c1, a2 = b2 = c2, and a3 > b3 > c3.


    Many thanks.




    Jen
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  2. #2
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    Quote Originally Posted by jenjen View Post
    Prove that A*B*C holds if we have a1 < b1 < c1, a2 = b2 = c2, and a3 > b3 > c3.
    Jen
    What does A*B*C mean?
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    Ohh, A * B * C means: B is between A and C
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    Quote Originally Posted by jenjen View Post


    1) Suppose that A, B, C are collinear points in R^3 whose coordinates are given by (a1, a2, a3), (b1, b2, b3), and (c1, c2, c3) respectively. Prove that A*B*C holds if we have a1 < b1 < c1, a2 = b2 = c2, and a3 > b3 > c3.
    I do not know how to prove something like this. But one of the undefined terms in Hilbert's geometry is "betweenness". However, it may be definied for numbers in \mathbb{R}^n The only thing I can say is that since it is one a line and the x coordinates are increasing means the largest ones contain the smaller one.
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  5. #5
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    Well, TPH this question may be beyond your experience.
    In any metric space ‘betweeness” is a well-defined concept.
    A*B*C means that d(A,B)+d(B,C)=d(A,C).

    Hilbert’s axioms are for a synthetic geometry.
    This question is not about synthetic geometry.
    In traditional mathematics, R^3 is not synthetic geometry.
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  6. #6
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    Quote Originally Posted by Plato View Post
    Well, TPH this question may be beyond your experience.
    In any metric space ‘betweeness” is a well-defined concept.
    A*B*C means that d(A,B)+d(B,C)=d(A,C).
    I actually understand you and a nice definition indeed.

    The metric here is the regular distance metric?
    d(\bold{u},\bold{v})=\sqrt{\sum_{k=1}^n (a_k-b_k)^2}
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    Ohh I am soo sorry, my definition wasn't so clear.
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