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Homework Help: Vector-valued function tangent

  1. Sep 13, 2010 #1
    1. The problem statement, all variables and given/known data

    If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t), show that the curve lies on the sphere with center at the origin.

    2. Relevant equations

    I know dot product might help:

    r(t) . r'(t) = 0

    and the equation of a sphere in 3-space:

    r2 = x2 + y2 + z2

    3. The attempt at a solution

    if I write out the components of the dot product...

    r(t) . r'(t) = fx(t)*fx'(t) + fy(t)*fy'(t) + fz(t)*fz'(t) = 0

    From there, I am not sure what to do, if that even is the right way to start.
     
  2. jcsd
  3. Sep 13, 2010 #2

    Mark44

    Staff: Mentor

    What if you integrate both sides of your last equation?
     
  4. Sep 13, 2010 #3
    Wow, how did you think of that?

    It seems to work. The one thing I need help with is integrating the right side of 0, I think it's my lack of calculus knowledge. Does it become a constant?
     
  5. Sep 13, 2010 #4

    Mark44

    Staff: Mentor

    I don't know - it just occurred to me because of those terms fx fx'.
    Yes.
     
  6. Sep 13, 2010 #5
    Thank you very much, just for completion's sake, I'll show the rest of the work:

    It is easier to write functions with different letters, so from before, fx(t) will now be f(t), fy(t) will now be g(t), and fz(t) is now h(t).

    [tex]\int f(t)df(t)[/tex] + [tex]\int g(t)dg(t)[/tex] + [tex]\int h(t)dh(t)[/tex] = [tex]\int 0dt[/tex]

    [tex]\stackrel{1}{2}[/tex] f2(t) + [tex]\stackrel{1}{2}[/tex] g2(t) + [tex]\stackrel{1}{2}[/tex] h2(t) = C

    f2(t) + g2(t) + h2(t) = r2 <-- form of a sphere.
     
    Last edited: Sep 13, 2010
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