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Some vector relating problems

  1. Mar 21, 2009 #1
    1. The problem statement, all variables and given/known data

    1. Suppose r(t)= (e^t * cost) i + (e^t * sint) j. Show that the angle between r and a never change. What is the angle.

    2. Find the equations for the osculating, normal, and rectifying planes of the curve r(t)=t i + t^2 j + t^3 k.

    3. Express the curvature of a twice differentiable curve r= f(theta) in the polar terms of r and its derivatives

    2. Relevant equations

    Kappa, Torsion, cross product, dot product,...

    3. The attempt at a solution

    1. So a is the acceleration. Thus, it is the 2nd derivative. So do I find the normal vector of r and a and then take their cross product to find the angle ? What will be the normal vector to r ??

    2. So is the osculating plane pretty much the curvature circle ? What is the rectifying plane ?

    3. I am kinda lost in the problem. How should I attack this ?
     
  2. jcsd
  3. Mar 21, 2009 #2

    gabbagabbahey

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    No, just take the second derivative. Do the unit vectors i and j ever change with time? If not, then the time-derivative of r is easy to find.
     
  4. Mar 21, 2009 #3
    I took the 2nd derivative. But I need to find the angle though ????
     
  5. Mar 21, 2009 #4

    gabbagabbahey

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    For any two vectors, u.v=||u||*||v||cos(theta)...where theta is the angle between them....so theta=___?
     
  6. Mar 21, 2009 #5
    Right. I got it but how do I prove it that it does not change ??
     
  7. Mar 21, 2009 #6

    gabbagabbahey

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    Well, what is d(theta)/dt?
     
  8. Mar 21, 2009 #7
    0 ??

    How about the 2nd problem ?
     
  9. Mar 21, 2009 #8
    I think I got the first problem solved. How about the next 2 ?
     
  10. Mar 22, 2009 #9
    For number 2, what is the equation of osculating and rectifying plane look like ? I know for the normal plane it is a(x-x0)+b(y-y0)+c(z-z0)=0

    Is the rectifying plane formed by the binormal vector and the unit tangent vector T ??

    Any suggestion for number 3 ?
     
  11. Mar 22, 2009 #10
    I have already solved number 2.

    Any suggestion for 3 ?
     
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