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Find the unit vector?

  1. Apr 14, 2010 #1
    1. The problem statement, all variables and given/known data

    Linear Algebra:

    For all the unit vectors u=[x,y,z]^T in R^3. Find the one for which the sum x+8y+2z is minimal. (u is a 3 x 1 vector)

    2. Relevant equations

    3. The attempt at a solution

    I tried working this with the least squares method...it wasn't right. I am probably overthinking this.

    Any help is appreciated! :)
  2. jcsd
  3. Apr 14, 2010 #2


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    Homework Helper

    all the unit vectors repressent a sphere of radius one

    consider the plane x+8y+2z = c for some arbitrary c, each c representing a a different plane

    you basically want to find the plane with the smallest c that still intersects the sphere (hint: which will be at only one point on the sphere... think directions)
  4. Apr 14, 2010 #3


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    Science Advisor

    The problem is simply to minimize x+ 8y+ 2z= 0 with the constraint [itex]x^2+ y^2+ z^2= 1[/itex].

    "Lagrange multiplier method" seems in order.
  5. Apr 14, 2010 #4


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    lagrange is good & will work, but i think if you just consider which direction leads to a single intersection of the plane and sphere you can skip a couple of steps, though all sama sama
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