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Polar to Cartesian Unit Vectors in 2D

  • Thread starter leahc
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  • #1
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Homework Statement


Solve for the unit vectors x-hat and y-hat in terms of r-hat and phi-hat.


Homework Equations


r-hat=cos(phi)x-hat+sin(phi)y-hat
phi-hat=cos(phi)y-hat-sin(phi)x-hat,


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.
 

Answers and Replies

  • #2
haruspex
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Let ##\hat{x} = \alpha\hat{r} + \beta\hat{\phi}##.
Compute ##\hat{x}.\hat{r}## etc.
 
  • #3
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I understand that, but how do I compute α and β?
 
  • #4
vela
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Haruspex already told you how. That's a dot product in case you didn't recognize it.
 
  • #5
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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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