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Linear Transformation in terms of Polar Coord.

  1. Jun 18, 2008 #1
    1. The problem statement, all variables and given/known data
    Let L(x) be the Linear operator in R^2 defined by
    L(x) = (x1 cos a - x2 sin a, x1 sin a + x2 cos a)^T

    Express x1, x2 & L(x) in terms of Polar coordinates.
    Describe geometrically the effects of the L.T.

    2. Relevant equations
    Well I know that:
    a = tan^-1 (x2 / x1)
    r = (x1^2 + x2^2)^1/2

    where a is the angle in both cases

    3. The attempt at a solution
    I know x1 & x2 in terms of polar coordinates is
    x1 = r cos a
    x2 = r sin a

    But Im not sure about L(x)... cos a & sin a stay the same in both coord?

    Thanks
     
  2. jcsd
  3. Jun 18, 2008 #2
    If they stay the same which I guess so... then by the properties of trigonometric functions...
    L(x) in polar coord will be
    [ (r cos 2a) , (r sin 2a) ] ?
     
  4. Jun 18, 2008 #3

    Defennder

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    Yeah that looks ok. Now observe what are the differences in the original vector in polar coordinates and the transformed vector. That should help geometrically.
     
  5. Jun 18, 2008 #4
    well... Obiously the a was doubled. But if the polar coordinates are of the form (r, a), How should I graph ( r cos 2a , r sin 2a )?

    :S
     
  6. Jun 18, 2008 #5
    is it a counter clock wise rotation by a, to whatever the line is?
     
  7. Jun 18, 2008 #6

    Dick

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    I really don't think you are supposed to take the angle the same for both x and L. L is a matrix which expresses a rotation by an angle a. If a general point p=(r*cos(t),r*sin(t)) then L(p) will rotate t->t+a, or maybe t->t-a, I'll let you figure out which.
     
  8. Jun 18, 2008 #7
    In my notes I have L(x)=(r*cos(a+t) , r*sin(a+t)). I though they would be the same so I reduced it to 2*a.

    So I can graph a point P as you described it, anywhere and then increase the angle by a to prove my point? No need to figure out where exactly r*cos(a) or r*sin(a) will actually be located?
     
  9. Jun 18, 2008 #8

    Dick

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    I don't think so. Just draw a picture that shows a linear transformation that rotates a vector by an angle a.
     
  10. Jun 18, 2008 #9
    You guys are the best! Thank you very much!!!!
     
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