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Homework Help: 2 questions on vector functions

  1. Jul 5, 2008 #1
    1. The problem statement, all variables and given/known data
    1) Find parametric equations for the line that is tangent to r(t) = (sin t)i + (t^2 - cos t)j + (e^t)k at the parameter value t = 0.

    2) For the equation r(t) = (cos t)i + (sin t)j and for t >= 0, is the particle's acceleration vector always orthogonal to its velocity vector?


    2. Relevant equations



    3. The attempt at a solution
    1) According to my text the tangent line to the curve r(t) = f(t)i + g(t)j + h(t)k is the line that passes through the point (f(t0), g(t0), h(t0)) parallel to v(t0), where t0 = 0 in this problem.
    I solved the velocity equation v(t) = (cos t)i + (2t + sin t)j + (e^t)k, and v(0) = i + k.
    This is the answer:
    x = t, y = -1, z = 1+t
    I found the point (f(0), g(0), t(0)) = (0, -1, 1), I'm not sure how to find the equation of a line that passes through this point and is parallel to another line? thanks

    2) For this one I just want to make sure, to find where the acceleration vector is orthogonal to its velocity vector I find where their dot products equal 0, right?
     
  2. jcsd
  3. Jul 6, 2008 #2

    Dick

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    Science Advisor
    Homework Helper

    The equation of a line passing through a point x(0) with instantaneous velocity v(0) (i.e. tangent line) is x(0)+v(0)*t. And yes, orthogonal means dot product zero.
     
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