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Derive relations for components by rotation of axes

  1. Feb 7, 2009 #1
    1. The problem statement, all variables and given/known data

    x1,x2) are the components of a 2 dimensional vector r when referred to cartesian axes along the directions i,j. derive the relations
    x1'= cosΘ x1+sinΘ x2
    x2'=-sinΘ x1+cosΘ x2
    for the components (x1',x2') or r referred to new axes i',j' obtained by a rotation of the axes through an angle Θ about the k direction

    2. Relevant equations

    3. The attempt at a solution
    i just wrote that r=x^2+y^2 which in this case would be x1^2+x2^2 and then accounted for rotation by multiplying by sin or cos theta and proving that x1^2+x2^2=x1'^2+x2'^2 but it dont think its sufficient
  2. jcsd
  3. Feb 7, 2009 #2


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    Homework Helper

    What's (1,0) rotated by an angle theta? Ditto for (0,1). To get those just draw a right triangle in the xy plane with angle theta at the origin. Use trig. Then (x1,x2)=x1*(1,0)+x2*(0,1).
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