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.

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

Quote:

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Originally Posted by jenjen

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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Since the questions concerns coordinate space, are you supposed to use vectors?

For question 1, you could treat a line through points and as a parametric equation Then to show the lines are distinct, show implies except when To do this takes a little work, but basically you treat the line L as a parametric equation and then show would imply one of two cases. Case 1 is and Case 2 is for and thus is on the line L. Either case contradicts the assumptions.