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Dot product of a vector with the derivative of its unit vector

  1. Mar 5, 2012 #1
    1. The problem statement, all variables and given/known data
    Let c(t) be a path of class C1. Suppose that ||c(t)||>0 for all t.
    Show that c(t) dot product with d/dt((c(t))/||c(t)||) =0 for every t.


    2. Relevant equations
    I am having trouble with the derivative of (c(t)/||c(t)||) and how to show that when its dotted with c(t) that it always equals zero.


    3. The attempt at a solution
    I know that c(t) dotted with c'(t) =0 when ||c(t)||= a constant but dont know how to implement this fact for this problem.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 5, 2012 #2

    Dick

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    If d(t)=c(t)/||c(t)||, what is ||d(t)||?
     
  4. Mar 5, 2012 #3

    jbunniii

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    What is the magnitude of c(t)/||c(t)||?
     
  5. Mar 5, 2012 #4
    would it be 1?
     
  6. Mar 5, 2012 #5

    Dick

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    Show why it is 1. Then you can drop the '?'.
     
  7. Mar 5, 2012 #6
    Ok i get that now. So if d(t)=1, which is a constant, then d(t) dotted with d'(t) =0. Since c(t)/||c(t)|| has the same direction as c(t), we can then plug c(t) in for d(t). Does that make sense to do?
     
  8. Mar 5, 2012 #7

    Dick

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    Makes sense to me.
     
  9. Mar 5, 2012 #8
    Awesome. Thanks for the help
     
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