A Bit Confused About Polar Basis Vectors

[MrBillyShears]

Let me say from the beginning I'm not talking about the non-coordinate unit vectors for polar coordinates. I'm talking about basis vectors. Let me just ask it as boldly as possible: how does one use these basis vectors in order to describe a vector? I know they are different at every point, so which point do you use? Is it completely arbitrary? Why is there different basis vectors at every point? And, I am new with this kind of stuff, so try to keep it as simple as possible in your explanation.
And also, if so, if you pick [itex]\vec{r}[/itex] to describe your vectors, would the "tails" of your vectors come from the origin, or from your point [itex]\vec{r}[/itex]? And, if someone could give an example with numbers, that would be great.

 

[Blazejr]

To really understand this concept properly one needs to learn some basic introduction to manifolds and tangent spaces. But let's first clarify on the underlying idea, without going into details. Those vectors you are referring to are in a way "glued to a point", and if you are going to imagine them as arrows, they would come from points in space, not always from origin. We call them vectors tangent to a point. Space of all vectors tangent to a given point is obviously linear space. Space of all such vectors (at any point) is not, because it's meaningless to add vectors with tails at different points! Therefore, for every point we have a tangent space. You can have different basis in those spaces (from now on I assume that we chose one point). We can have, for example, basis related to cartesian cordinates. Those basis vectors are of unit length and point in direction of coordinate axes. You can decompose any vector in terms of those basis vectors and get cartesian components. Make no mistake though - those components are not coordinates of a vector. Vector is not a point in the original space. Now you can also have basis related to polar coordinates. One vector pointing in the direction of growing [itex]r[/itex] etc., three vectors, each parallel to each other. There is important difference between those and cartesian vectors. If you go to different point in space. There direction of growing r is different! At point [itex](3,0,0)[/itex] in euclidean space cartesian coordinates of r vector would be just [itex](1,0,0)[/itex]. However, at point [itex](-3,0,0)[/itex] has cartesian components [itex](-1,0,0)[/itex]. Both point radially outwards and are of unit length, but what it means to point radially outward clearly depends on where you are.

 

[WWGD]

One issue is that, outside of R^n , there is rarely a natural isomorphism between vector spaces at different points. The differential quotient then takes tangent vectors at different points, and, like blazejr said, this difference --the whole expression-- is not well-defined. To "well-define" it , one uses connections, which are choices of vector-space isomorphisms between the tangent spaces.

 

[Fredrik]

Let me say from the beginning I'm not talking about the non-coordinate unit vectors for polar coordinates. I'm talking about basis vectors.

I don't understand what you're saying here. "Non-coordinate unit vectors" sounds like something that has nothing to do with the coordinate system. "for polar coordinates" sounds like the exact opposite. Then the second sentence suggests that the vectors you were talking about in the first sentence aren't basis vectors. Every linearly independent set with two vectors is a basis.

I'm still assuming that you're talking about these guys:
\begin{align}
&\hat r=(\cos\varphi,\sin\varphi)\\
&\hat\varphi =(-\sin\varphi,\cos\varphi)
\end{align} For all ##r,\varphi##, the vectors above are the ones that the polar coordidinate system associates with the tangent space at the point ##(r\cos\varphi,r\sin\varphi)##. Since that tangent space is an identical copy of the vector space that your ##\vec r## is an element of, you can also use these vectors as a basis for that space.

how does one use these basis vectors in order to describe a vector?

Same way you use any other basis.

I know they are different at every point, so which point do you use?

A point on a particle's trajectory at which you intend to calculate something, like that particle's centripetal acceleration.

Why is there different basis vectors at every point?

They're defined by
\begin{align}
\hat r=\frac{\frac{d\vec r}{dr}}{\left|\frac{d\vec r}{dr}\right|},\qquad \hat\varphi=\frac{\frac{d\vec r}{d\varphi}}{\left|\frac{d\vec r}{d\varphi}\right|}
\end{align} The right-hand sides have different values at each point. If you don't find this useful, then you can use some other basis. But you will certainly find these bases useful when you describe circular motion, where the particle's position and velocity are respectively ##r\hat r## and ##r\dot\varphi\hat\varphi##, at every point.

if you pick [itex]\vec{r}[/itex] to describe your vectors, would the "tails" of your vectors come from the origin, or from your point [itex]\vec{r}[/itex]?

The position vector should be drawn from the origin, but the velocity vector from the particle's location. It makes sense to think of the position vector ##\vec r## as an element of your original copy of ##\mathbb R^2## ("the configuration space"), and the velocity vector as an element of a different copy of ##\mathbb R^2## ("the tangent space at ##\vec r##") with its origin attached to the point ##\vec r##. When you're dealing with a significantly less trivial manifold than ##\mathbb R^2##, you pretty much have to think this way.

 

[blue_raver22]

Firstly, it's important to understand that basis vectors are a fundamental concept in linear algebra and are used to represent vector spaces. In the case of polar coordinates, the basis vectors are not the same as the non-coordinate unit vectors, as you mentioned. These basis vectors are used to describe vectors in a polar coordinate system, which is a different way of representing points in space.

In order to describe a vector using basis vectors, you need to understand that the basis vectors are defined at every point in space. This means that at each point, there are two basis vectors - one for the radial direction and one for the tangential direction. The radial basis vector points towards the center of the coordinate system, while the tangential basis vector points in the direction of increasing angle.

To use these basis vectors to describe a vector, you need to express the vector in terms of these basis vectors. This is done by finding the scalar components of the vector in the radial and tangential directions, and then multiplying them by their respective basis vectors. This will give you the vector expressed in terms of the basis vectors at a specific point.

It's important to note that the choice of point is not arbitrary. The basis vectors are specific to each point in space and are dependent on the coordinate system being used. So if you change the point, the basis vectors will also change.

For example, let's say we have a vector in polar coordinates with a magnitude of 5 and an angle of 30 degrees. To describe this vector using basis vectors, we would first need to find the scalar components. The radial component would be 5*cos(30) = 4.33 and the tangential component would be 5*sin(30) = 2.5. Then, we would multiply these components by their respective basis vectors at the chosen point. If we choose the point (2, 30 degrees), the radial basis vector would be (1, 0) and the tangential basis vector would be (0, 1). Therefore, the vector would be expressed as (4.33, 2.5) in terms of the basis vectors at that point.

To address your second question, if you choose the vector \vec{r} to describe your vectors, the "tails" of your vectors would come from the origin. This is because \vec{r} represents the position vector of a point in polar coordinates, and the origin is the reference point for this coordinate