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Find angle to eliminate cross product term

  1. Nov 22, 2009 #1
    1. The problem statement, all variables and given/known data

    If you make the change of variables:

    x= ucos([tex]\theta[/tex])-vsin([tex]\theta[/tex])
    y= usin([tex]\theta[/tex])+vcos([tex]\theta[/tex])


    where the angle 0 [tex]\leq[/tex] [tex]\theta[/tex] <[tex]\pi/2[/tex] is chosen in order to eliminate the cross product term in:

    4x2+8xy+6y2=30

    What is the angle you would use?

    2. Relevant equations



    3. The attempt at a solution
    I have no idea. Would you find when the 8xy term (in terms of the new variables) is zero (solve for theta)?
     
  2. jcsd
  3. Nov 22, 2009 #2

    HallsofIvy

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    Not just the xy term. Each of the three terms will have "uv" in them.

    Replace x with [itex]u cos(\theta)- v sin(\theta)[/itex] and y with [itex]u sin(\theta)+ v cos(\theta)[/itex] in [itex]4x^2+ 8xy+ 6 y^2[/itex]. There will be "uv" terms in all three of those terms. Add then up and choose [itex]\theta[/itex] to make the coefficient of uv equal to 0.
     
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