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Determine which points are on a given line

by RoRi

Let L be a line containing the points P = (-3,1) and Q = (1,1). Determine which of the following points are also on L.

  1. (0,0);
  2. (0,1);
  3. (1,2);
  4. (2,1);
  5. (-2,1);
By Theorem 13.4 (page 474 of Apostol)we know that given two points P, Q there is a unique line L between them which is the set of points

    \[ L = L(P; P-Q) = \{ P + t(Q-P) \} = \{ (-3,1) + t(4,0) \}. \]

So, the set of points on L are all points of the form (4t-3, 1) for t \in \mathbb{R}.

  1. The point (0,0) is not on L since there is no t \in \mathbb{R} such that (4t-3,1)= (0,0).
  2. The point (0,1) is on L since if we take t = -\frac{3}{4} then (4t-3,1) = (0,1).
  3. The point (1,2) is not on L since (1,2) \neq (4t-3,1) for any t.
  4. The point (2,1) is on L since if we take t = \frac{5}{4} then (4t-3,1) = (2,1).
  5. The point (-2,1) is on L since if we take t = \frac{1}{4} then (4t-3, 1) = (-2,1).

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