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Homework Help: Find a normal vector to a unit sphere using cartesian coordinates

  1. Oct 8, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider a unit sphere centered at the origin. In terms of the Cartesian unit vectors i, j and k, find the unit normal vector on the surface

    2. Relevant equations
    A dot B = AB cos(theta)
    A cross B = AB (normal vector) sin(theta)
    Unit sphere radius = 1

    3. The attempt at a solution
    Isn't any direction a normal vector?
    i x j = + k
    j x i = - k
  2. jcsd
  3. Oct 8, 2015 #2


    Staff: Mentor

    Any nonzero vector would be a normal at some point on the surface of the sphere. My guess is that you should take an arbitrary point P(x, y, z) on the surface, and find the normal to it. If that's what is wanted in the problem, it could have been written more clearly.
  4. Oct 8, 2015 #3
    Thank you, that would make a bit more sense.
  5. Oct 9, 2015 #4


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    Science Advisor

    The unit sphere is of the form [itex]x^2+ y^2+ z^2= 1[/itex]. You can think of that as a 'Level Surface" of the function [itex]F(x, y, z)= x^2+ y^2+ z^2[/itex] and use the fact that the gradient of such a function, [itex]\nabla F[/itex], is always normal to level surfaces.
  6. Oct 9, 2015 #5


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    Science Advisor
    Homework Helper
    Gold Member

    Or think about what direction a position vector to a point on the sphere has.
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