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Calculus Question: Gradients & Normals

  1. Oct 19, 2004 #1
    Geometry Question: Gradients & Normal lines & planes

    This is a question from a past midterm that I'd appreciate some help with. It deals with the gradient as a normal. I'm not having trouble actually obtaining the gradient, but I am having trouble with some of the geometry involved, so any help would be appreciated!

    Consider the function:

    [tex]y = \sqrt{x^2 + z^2}[/tex]

    Give the equation for 2 planes whose intersection is the normal line to this surface at [tex](1,4,\sqrt{15})[/tex].

    I found the value: [tex]\nabla f(1,4,\sqrt{15}) = (1/4, -1, \sqrt{15}/4)[/tex].

    And the equation for the normal line is:

    [tex]r(t) = (1,4,\sqrt{15}) + t(1/4, -1, \sqrt{15}/4)[/tex]

    My question is: How do I find two planes that intersect in this line?

    I think I should parametrize the variables, so

    x = 1 + 1/4t
    y = 4 - t
    [tex]z = \sqrt{15} + \sqrt{15}/4 * t[/tex]

    But that's where I get lost. Can someone just point me in the right direction in terms of what equations to set up?

    Thanks!
     
    Last edited: Oct 19, 2004
  2. jcsd
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