relationship of polar unit vectors to rectangular I'm looking at page 16 of Fleisch's Student's Guide to Vectors and Tensors.  The author is talking about the relationship between the unit vector in 2D rectangular vs polar coordinate systems.  They give these equations:
\begin{align}\hat{r} &= \cos(\theta)\hat{i} + \sin(\theta)\hat{j}\\
\hat{\theta} &= -\sin(\theta)\hat{i} + \cos(\theta)\hat{j}\end{align}
I'm just not getting it.  I understand how, in rectangular coordinates, $x = r \cos(\theta)$, but the unit vectors are just not computing.

I too got stuck there, and found the answer in a YouTube video at:  http://www.youtube.com/watch?v=WwQTTJdAJP8. 
The difficulty I had was confusing the geometry of polar coordinates, where x = rcos(theta) etc, with the vector situation, where the radius vector is resolved into its components in the x and y directions, so, applying the formulae for expressing a vector in terms of its components to the case of the polar unit radial vector, we get the equation quoted: r^=cos(θ)i^+sin(θ)j^. Attached is an image from the video that gives the geometry of this equation.

An image from the final moment of the video.
