I was learning about torque, angular momentum etc. using polar coordinates. In the lecture, I learned that the unit vectors are $\hat{}$ and $\hat{r}$, where is the angle and r is the distance from the origin (i.e. the radius).
I can understand that r has a direction which is along the radius, therefore it is a vector. However, I don't understand why can by a vector: is an angle. How can an angle have a direction?
Thank you.
On
After some time reading your answers, thinking about and discussing the concept of $\hat{}$. I seem to have the explanation:
Here I am talking about the coordinates in 2-D (In 3-D and higher dimensions, the logic is exactly the same). In order to give full information of the coordinates, I need two components and they better be orthogonal to each other because, if they are not perpendicular to each other, when I do the problems, I still have to resolve one of the components to make them perpendicular to each other so that they don't affect each other. The two components can be defined in a Cartesian way, i.e. using the conventional $\hat{x}$ and $\hat{y}$ or $\hat{i}$ and $\hat{j}$, or whatever notations I want.
And similarly, I can give coordinates this in a polar way, in which one component is along $r$, i.e. $\hat{r}$. Again, to give the full information of the coordinates, I would better use a perpendicular component to $\hat{r}$, i.e. $\hat{}$. In other words, what I am suggesting here is that, in my opinion, $\hat{}$ is not a unit vector for an angle, it is the perpendicular component to $\hat{r}$ that, together with $\hat{r}$, tell us the exact location of the coordinates. The relationship between $\hat{r}$ and $\hat{}$ are essentially exactly the same as that between $\hat{x}$ and $\hat{y}$. Therefore, $\hat{}$ causes my confusion because of its notation, if I want, I can use other notations such as $\hat{s}$ (as s is the next letter after r in the alphabet).
Do you agree with me? Thank you!
On
As I think you have gathered by now, $\hat r$ is not $r$ turned into a vector and $\hat \varphi$ is not $\varphi$ turned into a vector.
In fact, there is no such thing as "the" vector $\hat r$ or "the" vector $\hat \varphi$ in polar coordinates. When we say that the polar coordinates of a point $P$ are $(r,\varphi) = \left(3, \frac\pi2\right)$, there are no vectors, call them $\hat r$ and $\hat \varphi$ or anything else, that we can multiply by $r$ and $\varphi$ in order to say anything useful about the location of the point $P$.
In my experience, vectors called $\hat r$ and $\hat \varphi$ (or the same vectors called by other names) occur in a context where we have already fixed our attention on a particular point $P$ in the plane, and the coordinates of that point, $(r, \varphi)$, are not going to change as long as we are dealing with $\hat r$ and $\hat \varphi$. Then the parameter $\varphi$ is the angle from the positive $x$-axis in which the vector $\hat r$ (not $\hat\varphi$) is pointing, and $\hat\varphi$ is at a right angle counterclockwise from $\hat r$. That is, $\hat\varphi$ points in the direction $\varphi + \frac\pi2$.
It might be clearer if people would write $\hat r_P$ and $\hat\varphi_P$ in order to remind us about the point in the plane that determined the definitions of these vectors, or at least $\hat r_\varphi$ and $\hat\varphi_\varphi$ to acknowledge that the vectors depend on the coordinate $\varphi$ at $P$. But I don't think I have ever seen these vectors named like that. People expect you to know from context that the vectors are defined relative to a particular point and to remember what that point is.
We can see that once we have selected the point $P$ with polar coordinates $(r, \varphi)$ at which we want to define the vectors $\hat r$ and $\hat \varphi$, the position vector of $P$ itself is $r\hat r$. In my view, this is a useless, question-begging observation. Unlike an expression such as $x\hat x + y\hat y$, you don't use $r\hat r$ to find or identify a point in the plane; you come up with $r\hat r$ after you have already identified a point in the plane, or at least the ray out of the origin on which that point lies. If you were to choose two random points $P_1 = (r_1,\varphi_1)$ and $P_2 = (r_1,\varphi_2)$ in the plane, with $r_1 > 0$ and $r_2 > 0$, you could not write their positions as $r_1\hat r$ and $r_2\hat r$ unless $P_1$ and $P_2$ happen to be collinear with the origin and both are on the same side of the origin.
What $\hat r$ and $\hat \varphi$ actually provide is a useful Cartesian coordinate system for looking at things that are happening at some identified point in the plane. For example, if we have a satellite orbiting a planet situated at the origin of the plane, the planet's radial component of velocity is in the direction of $\hat r$ and its tangential component of velocity is in the direction of $\hat\varphi$. We can work out forces, accelerations, and so forth all in this coordinate system, and because the coordinate system is Cartesian, it is easy to break vectors down into components and to add vectors componentwise.
So the things I would keep in mind when I see $\hat r$ and $\hat\varphi$ in a physics context are:
Whoever is using these symbols has almost certainly already identified a particular point relative to which these vectors are defined.
The vectors $\hat r$ and $\hat\varphi$ provide a useful "local" Cartesian coordinate system at that particular point, conveniently oriented with $\hat r$ pointing directly away from the origin, which usually has some physical significance.
You can do a lot of calculation in the "local" Cartesian coordinate system without referring to polar coordinates at all.
Once you have finished your calculations in the "local" system, you can take the conclusions back to the polar coordinate system.
Your confusion takes its origin in the notation. Actually, $\hat{\phi}$ is not an angle. Indeed, $\{\hat{r},\hat{\phi}\}$ forms a basis of $\Bbb{R}^2$, made of orthogonal unit vectors, like the canonical cartesian basis $\{\hat{x},\hat{y}\}$. However, this notation tends to conflate the basis vectors with the associated coordinates.
Let's take a point $P = (x,y) = (r,\phi)$ in the plane with cartesian and polar coordinates. Then, $\vec{OP} = x\hat{x} + y\hat{y} = r\hat{r}$. For instance, the abscissa $x$ may be interpreted as a length and a horizontal position, but $\hat{x}$ is only a unit vector and nothing else. The same goes for polar coordinates. It is also to be noted that the polar coordinates are defined in such a way that points of the plane are reached radially, so that $\hat{\phi}$ is not even needed.
Where has the angular coordinate $\phi$ defining the point $P$ gone ? Actually, it is "hidden" in $\hat{r}$. Indeed, coming back to the original cartesian coordinates, one has : $$ \vec{OP} = r\hat{r} = \sqrt{x^2+y^2} \left(\cos\phi\,\hat{x} + \sin\phi\,\hat{y}\right) $$
So, what's the use of $\hat{\phi}$ ? It is only here to complete the basis by representing a (counter-clockwise) direction orthogonal to $\hat{r}$, namely $\hat{\phi} = -\sin\phi\,\hat{x} + \cos\phi\,\hat{y}$ in consequence $-$ in fact, the polar basis $\{\hat{r},\hat{\phi}\}$ is obtained by rotating the cartesian basis $\{\hat{x},\hat{y}\}$ by an angle $\phi$.
This misleading mixing may be avoided by renaming the basis vectors by $\hat{e}_r$ and $\hat{e}_\phi$ or even $\hat{e}_1,\hat{e}_2$.