I think I have a conceptual issue with vectors in curvilinear spaces. For simplicity's sake I'll describe the problem in polar, but it also applies to cylindrical and spherical.
Let's say we have a point $P$ that is the terminus of a position vector $\vec r$. In Cartesian coordinates, $\vec r =\langle P_x, P_y\rangle = P_x\hat i +P_y \hat j$ which is nice and simple and intuitive. Alternatively, we could describe its position by its distance $r$ from the origin and the angle between $\vec r$ and the $x$-axis. And it doesn't take much geometry to show that now $\vec r=rcos\theta\hat i+rsin\theta\hat j$ where $r=\sqrt{P_x^2+P_y^2}$. Then, since $cos\theta\hat i+sin\theta\hat j$ is also a unit vector and it points in the same diection as $\vec r$, we call it $\hat r$ and $\vec r = r\hat r$. Beautiful.
I have also seen that $\vec r$ can be defined in polar as $\langle P_r, P_\theta \rangle = P_r\hat r+P_\theta \hat \theta$ where $\hat \theta=-sin\theta \hat i+cos\theta \hat j$ which has the interpretation of just being $\hat j$ but rotated by $\theta$, as well as being the "direction of increasing $\theta$." That much makes sense, but what's weird and where I get confused is that we already defined $\vec r$ in polar and there was no need for $\hat \theta$. Intuitively, since $\hat r$ points towards $P$ and $\hat \theta$ is perpendicular to $\hat r$, there shouldn't be any $\hat \theta$ component in $\vec r$. Heck, I can even prove it (well, not actually "prove" since I know something has to be wrong): $$\vec r=P_x\hat i+P_y\hat j=P_r\hat r+P_\theta \hat \theta$$ $$rcos\theta\hat i+rsin\theta\hat j=P_r(cos\theta\hat i+sin\theta\hat j)+P_\theta(-sin\theta \hat i+cos\theta \hat j)$$Which, pairing up the $\hat i$ and $\hat j$ terms gives $$rcos\theta=P_rcos\theta-P_\theta sin\theta$$ $$rsin\theta =P_rsin\theta +P_\theta cos\theta$$ And by matching coefficients you can see that $P_\theta$ has to equal 0.
Now, obviously $\hat \theta$exists for a reason. So what I'm assuming is that this problem is conceptual and not arithmetic. What I'm assuming is that the entire concept of a vector as being a straight arrow just falls apart in non-rectangular coordinate systems, that the algebraic issues occur when I set $\vec r_{cart}=\vec r_{polar}$. I'm hoping that someone out there can give me a decent conceptual framework for what $\hat r$ and $\hat \theta$ really are, or at the very least prove my confirm my assumptions.
Many thanks in advance.
(Hooray for my first StackExchange post!)