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Apostle, chapter 14 problems: Vector Calculus

  1. Oct 22, 2011 #1
    1. The problem statement, all variables and given/known data
    Problem 1: Two fixed unit vectors A and B make an angle θ with each other, where 0 < θ < π. A particle moves in a space curve in such a way that its position vector r(t) and velocity v(t) are related by the equation v(t) = A x r(t). If r(0) = B, prove that the curve has constant curvature and compute this curvature in terms of θ.

    Problem 2: A point moves in space according to the vector equation r(t) = 4cos(t)i + 4sin(t)j + 4cos(t)k. Show this path is an ellipse.


    2. Relevant equations


    k = | a x v |/|v|^3


    3. The attempt at a solution


    1: I've already figured out the motion must move in a plane because r(t) x v(t) = A and is constant, but I don't know where to go from there.The answer is k = 1/|B|*sin(θ).

    2: I put x = 4cos(t), y = 4sin(t), and z = 4cos(t), and deduce xz + y^2 = 16 but this is not a ellipse. Where am I going wrong?
     
  2. jcsd
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