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.

The (endpoint of the) vector $(\cos\theta,\,\sin\theta)$ is on the unit circle, exactly at angle $\theta$ (if angles are measured from the $x$-axis, towards the $y$-axis). So, in polar form, we can say $r=1$ and $\theta=\theta$.
The other one, $(-\sin\theta,\,\cos\theta)$ is its rotated version, by $+90^\circ$. So, this has $\hat r=1$ and $\hat\theta=\theta+90^\circ$.
