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Math Help - points proof

  1. #1
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    points proof

    1) Let L be a line in coordinate 2- space R^2 or 3- space R^3, let x be a point not on L, and let p1,...pn be points on L. Prove that the lines xp1,...xpn are distinct. Why does this imply that R^2 and R^3 contain infinitely many lines?



    2) Suppose that L1,...Ln are lines in coordinate 2- space R^2 or 3- space R^3. Prove that there is a point q which does not lie on any of these lines. (hint: Take a line M which is different from each of L1,...Ln; for each j we know that M and Lj have at most one point in common, but we also know how M has infinitely many points.)



    Thanks a whole lot.
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  2. #2
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    Hi ThePerfectHacker,

    Thank you so much for trying to help me out but I don't think that is how I am suppose to prove it. Still, thanks a lot.



    Jen
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  3. #3
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    Quote Originally Posted by jenjen View Post
    1) Let L be a line in coordinate 2- space R^2 or 3- space R^3, let x be a point not on L, and let p1,...pn be points on L. Prove that the lines xp1,...xpn are distinct. Why does this imply that R^2 and R^3 contain infinitely many lines?



    2) Suppose that L1,...Ln are lines in coordinate 2- space R^2 or 3- space R^3. Prove that there is a point q which does not lie on any of these lines. (hint: Take a line M which is different from each of L1,...Ln; for each j we know that M and Lj have at most one point in common, but we also know how M has infinitely many points.)



    Thanks a whole lot.
    Since the questions concerns coordinate space, are you supposed to use vectors?

    For question 1, you could treat a line through points x and p_n as a parametric equation v_n (t) = (p_n - x)t + x. Then to show the lines are distinct, show p_i \ne p_j implies v_i (t_i) \ne v_j (t_j) except when t_i = t_j = 0. To do this takes a little work, but basically you treat the line L as a parametric equation w (t) = (p_i - p_j)t + p_j and then show v_i (t_i) = v_j (t_j) would imply one of two cases. Case 1 is t_i = t_j and p_i = p_j . Case 2 is x = w(t) for t = t_i/(t_i-t_j) and thus x is on the line L. Either case contradicts the assumptions.
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