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Calc III: Vector parallel to the line of intersection

  1. Mar 9, 2008 #1
    1. The problem statement, all variables and given/known data
    Hi there,

    Two points on the sphere S of radius 1 have spherical coordinates P: \phi = 0.35 \pi, \theta = 0.8 \pi and Q: \phi = 0.7 \pi, \theta = 0.75 \pi. Find a vector parallel to the line of intersection of the tangent planes to S at the points P and Q.

    2. The attempt at a solution
    First, I converted spherical to cartesian coordinates.
    P (-.72083942, .26684892, -.80901699)
    Q (-.5720614, -.41562694, -.70710678)

    Then, I found two tangent planes at P and Q. To do this, I found the gradient of the general sphere equation x^2 + y^2 +z^2 = 1. And then the general equation of the tangent plane for each point.

    Tangent P = -1.44167x + .533698y - 1.161803z -2.4906521

    Tangent Q = -1.1441228x - .83125388y - 1.4142136z -2

    3. Relevant equations
    I'm lost on what to do after that though. I talked to a tutor about this question and he said to set the two equation so that they equal z. Then add them together to get a single equation of a line. And then use parametrics since it's asking for a vector parallel. Does this sound right?
  2. jcsd
  3. Mar 10, 2008 #2
    /nudges to the top
    Wake up your monday morning with some hearty, delicious calculus! ;)
  4. Mar 10, 2008 #3


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    Homework Helper

    That looks very complicated. :frown:

    The tangent planes at P and Q are perpendicular to OP and to OQ.

    So they're simply r.P = 0 and r.Q = 0, and they intersect in the line r.P = r.Q = 0.

    Try that! :smile:
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