Polar to Cartesian Unit Vectors in 2D

[leahc]

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.

 

[haruspex]

Let $\hat{x} = \alpha\hat{r} + \beta\hat{\phi}$.
Compute $\hat{x}.\hat{r}$ etc.

 

[leahc]

I understand that, but how do I compute α and β?

 

[vela]

Haruspex already told you how. That's a dot product in case you didn't recognize it.

 

[NemoReally]

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?