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Polar to Cartesian Unit Vectors in 2D

  1. Jan 27, 2013 #1
    1. The problem statement, all variables and given/known data
    Solve for the unit vectors x-hat and y-hat in terms of r-hat and phi-hat.


    2. Relevant equations
    r-hat=cos(phi)x-hat+sin(phi)y-hat
    phi-hat=cos(phi)y-hat-sin(phi)x-hat,


    3. The attempt at a solution
    I have been working on this for a really long time, and I keep getting a really complicated expression for x-hat, with everything over cos(phi)^2. That seems wrong, and I can't figure out how to solve it from only those two given equations.
     
  2. jcsd
  3. Jan 28, 2013 #2

    haruspex

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    Let ##\hat{x} = \alpha\hat{r} + \beta\hat{\phi}##.
    Compute ##\hat{x}.\hat{r}## etc.
     
  4. Jan 28, 2013 #3
    I understand that, but how do I compute α and β?
     
  5. Jan 28, 2013 #4

    vela

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    Haruspex already told you how. That's a dot product in case you didn't recognize it.
     
  6. Jan 28, 2013 #5
    Where did you get those equations? Why did you choose them? Take one example of an x-y unit vector and calculate r and phi - plotting it might help. Then take a different example and do likewise. Notice anything?
     
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