Calc. 3 Determining whether points lie in a straight line

  1. Hey, how do I determine whether or not points lie in a straight line? Is there a symbolic approach to determining so? Or do I need to spatially visualize it?

    For instance,

    A(0,-5,5), B(1,-2,4), C(3,4,2) does lie in a straight line according to my book. Thanks!
  2. jcsd
  3. Do you know any linear algebra?
    If [0, -5, -5], [1, -2, 4], and [3, 4, 2] all lie in a one dimensional subspace, what does that say about their linear independence?
  4. What is the vector going from A to B (call it v)? If C lies on this line than A-C = tv. Is this true?
  5. HallsofIvy

    HallsofIvy 41,268
    Staff Emeritus
    Science Advisor

    Three points can lie on a straight line and NOT lie in a one dimensional subspace so that is NOT sufficient. Only lines that include the origin form a subspace.
  6. Hey d.tran103 and welcome to the forums.

    What you can do is calculate the direction vector of each combination of lines by calculating vectors for lines AB, BC, and AC. Then plug these into a matrix and show whether all of these are linearly dependent.

    You can't just plug the raw points in because as HallsOfIvy said, this only works when all lines go through the origin, but if you look at direction vectors of the lines then this is a different thing and you can apply the techniques used in linear algebra.

    If they all lie on the same line, then your reduce system will have 1 row of values and the rest will be all 0 elements. If you don't get this situation, then you know that there is at least one other independent characteristic.
  7. The way I think of this is that you consider one of the points as the origin. Then regular linear algebra applies.
  8. This is equivalent to what I said above.
  9. ^yes. I am not sure why this matters.
  10. HallsofIvy

    HallsofIvy 41,268
    Staff Emeritus
    Science Advisor

    If you are given the three points to be "A(0,-5,5), B(1,-2,4), C(3,4,2)", then the coordinate system is given and you cannot "take one of them to be the origin".

    The simplest thing to do is to form the vectors AB= <1- 0, -2- (-5), 4- 5>= <1, 3, 1> and AC== <3- 1, 4- (-2), 2- 4>= <2, 6, -2>. The three points lie on a single line if and only if one of those two vectors is a multiple of the other.

    Of course, that is equivalent to translating the three points so that A is translated to the origin but that is not just "taking one point to be the origin".
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