Unit Vector polar in terms of cartesian

 Homework Sci Advisor HW Helper Thanks P: 12,463 How did you do the first bit? What would be the analogous method for the second bit? You are not asked to prove it, just write it down. note: $\text{\hat{a}} \rightarrow \hat{a}$ ... rather than a{hat}. (welcome to PF)
 P: 260 I'm not sure if this will help you, but the general form of the transformation of a basis vector is: $$\vec{e}_{\bar{\nu}}=\sum_{\mu=1}^n \frac{ \partial x^\mu }{ \partial x^{\bar{\nu}}}\vec{e_\mu}$$ where n is the number of dimensions (in this case two). xμ represents the Cartesian coordinates x and y (i.e. x1=x, x2=y). xν (with a bar over it - this distinguishes between Cartesian and polar coordinates) represents the polar coordinates r and θ. What you need to do is differentiate the Cartesian coordinates x and y with respect to r and θ (i.e. dx/dr, dx/dθ, dy/dr, and dy/dθ). When you sum the Cartesian basis vectors e1=(1,0) and e2=(0,1) times the appropriate values, you'll get basis vectors for r and θ.
 P: 29 Unit Vector polar in terms of cartesian Could someone give me a hint on the first part of this? Because I can derive it - that is just simple trigonometry - but I can't figure out how to concretely prove that $\hat{r}= \hat{x}cosθ + \hat{y}sinθ$ Edit: I'm thinking illustrate that $\vec{r} = r\hat{r}$ in polar and then showing that in Cartesian $\vec{r} = \hat{x}cos\phi +\hat{y}sin\phi$ Edit2: Nope, I'm confused again.. I think elfmotat is correct, but I don't quite understand his explanation. Edit3: nevermind - I got it.