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Proving derivatives of a parametrized line are parallel

  1. Jun 22, 2014 #1
    1. The problem statement, all variables and given/known data

    Show that if σ(t) for (t in I) is a parametrization of a line, then σ''(t) is parallel to σ'(t).

    2. Relevant equations



    3. The attempt at a solution

    I thought that if σ(t) is a parametrization of a line then it could be expressed as σ(t) = vt + a, but then σ'(t) = v and σ''(t) = 0. Can two vectors be parallel? Is it because the distance between them is constant? Or did I make a mistake earlier on?
     
  2. jcsd
  3. Jun 23, 2014 #2

    LCKurtz

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    That would be the equation of a line where the point ##\sigma(t)## moves with constant velocity. But what about something of the form ##\sigma(t) = f(t)\vec D + \vec a##? Wouldn't that give a straight line too?
     
  4. Jun 23, 2014 #3
    Ah, thank you! But both ##\vec{D}## and ##\vec{a}## are still just regular vectors, right?
     
  5. Jun 23, 2014 #4

    LCKurtz

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    Yes. ##\vec D## is a direction vector and ##\vec a## is a position vector to a point on the line. Both vectors are constants.
     
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