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Homework Help: Planes perpendicular to vectors

  1. Feb 7, 2010 #1
    a)Find the equation of the tangent plane to the graph z = x2 + 4y2 at the point (a, b, a2 + 4b2)

    b) For what values of a and b is the tangent plane perpendicular to the vector (3, -4, 2)?

    I figured you have to use the tangent plane equation z - zo = zx(a,b)(x-a) + zy(a,b)(y-b). This gave me the equation

    Z = 2ax + 8by - a2 - 4b2

    For b, I think the dot product could be used if I set it to zero. But how would I go about solving for a and b? I think I may just be getting tired and a bit sloppy, because I know this is already stuff I have gone over, the only new thing is solving for a and b.

    Any direction on this would be great, I'm just having trouble visualizing this problem.
     
    Last edited: Feb 7, 2010
  2. jcsd
  3. Feb 8, 2010 #2
    I couldn't get to sleep because I kept on thinking this problem through. As far as I can see, if the plane's norm is the same as the vector <3, -4, 2>, then it must be perpendicular to it. I am not sure how to make the norm on the plane into that though based off of just of values of a and b. Does anyone have any suggestions?
     
  4. Feb 8, 2010 #3
    Doesn't have to be the same, just parallel, i.e. costheta=1 or -1.

    Find the equation of the norm of the plane first.
     
  5. Feb 8, 2010 #4
    The norm of my plane is influenced by a and b. This is what I have:

    2a (X - a) + 8b (y - b) - z + a^2 + 4b^2 = 0

    So if the vector is <3, -4, 2>, I could put the norm of my plane as <-3/2, 2, - 1> since that would be parallel? Hence a = -3/4 and b = 1/4 ?
     
  6. Feb 8, 2010 #5

    Hm, could you post the equation of the normal?
     
  7. Feb 8, 2010 #6
    Sorry but I'm not sure what you mean by the equation of the normal. Do you mean the vector equation of the tangent plane?

    <-3/2, 2, -1>[tex]\cdot[/tex]<x + 3/4, y - 1/4, z> = 0?

    Although I do still have those other numbers a2 + 4b2. Would that influence the normal of my plane still? I am tired and will probably think about this better later. Thanks for the help so far!
     
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