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From other responses on this and other platforms I have finally gathered that $\vec p=a_ρ\vec e_ρ+a_φ\vec e_φ$ does in fact not appear to represent a point in polar coordinates the way $a_x\vec e_x+a_y\vec e_y$ would in kartesian coordinates.

What however I still am strugling to understand is what then the meaning of this notation - especially the second summand is?

To represent a point the first term $a_ρ\vec e_ρ$ should be sufficient, provided both the basis vector and coefficient are known. Maybe $\vec p=a_ρ\vec e_ρ+a_φ\vec e_φ$ could then somehow lead to the relevant $ρ$ and $φ$ without knowing the basis vectors? However in that case the problem of $φ$ changing after the first term returns - i.e. one is using summation of polar basis vectors to represent a point again?

Hopefully my confusion is at least understandable and I haven't failed to express myself, thanks in advance for responading and for reading.

Robbe
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  • You are describing coordinate vectors. These are not basis vectors. Linear combinations of basis vectors describe other vectors in a space of vectors. Coordinate vectors are used to describe positions of points in a geometric or physical space. – John Douma Oct 02 '24 at 21:26
  • @JohnDouma You're right ... however that doesn't really help me unfortunately (maybe it should, would you care to explain?). It shouldn't really matter if one is considering the "coordinate vector" from the origin to (r,φ) or the "vector-vector" (r,φ): It still is representable by only $a_ρ\vec e_ρ$ - so what's the meaning of the second term in a vector described as a sum of both basis vectors? – Robbe Oct 02 '24 at 21:45
  • The coordinate vector has $\theta$ built into it because $\hat r=\cos{\theta}\hat i + \sin{\theta}\hat j$. Take a look at this. – John Douma Oct 02 '24 at 22:02
  • Very good read! Thanks for sharing - the "each θ is a different coordinate system" idea is a great way to represent it actually! However ... I'm still not absolutely sure about my original question (although this might have helped express myself better): If there were a question of the kind: "Given the coordinate vector $\vec p=r\vec e_r+θ\vec e_θ$" - what does that mean? $r\vec e_r$ is sufficient to describe the vector - the second term just adds some random offset? Is it just wrong notation to write down the θ in a simple manner? Or is the second coefficient not really θ but related somehw? – Robbe Oct 02 '24 at 22:21
  • The $\hat\theta$ component comes into effect when the particle moves. For example, if a particle's position $\vec p$ is $\vec p = r\hat r$, then $\frac{d\vec p}{dt}=r\frac{d\hat r}{d\theta}\frac{d\theta}{dt}+\frac{dr}{dt}\hat r=r\frac{d\theta}{dt}\hat\theta+\frac{dr}{dt}\hat r$. In other words, the velocity is the radial speed ($\frac{dr}{dt}$)in the radial direction plus the rotational speed ($r\frac{d\theta}{dt}$) in the direction of counter-clockwise rotation. – John Douma Oct 02 '24 at 22:48
  • Maybe the concrete example at the bottom of this answer helps further. – Kurt G. Oct 03 '24 at 05:40

1 Answers1

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Vectors are not points.

It so happens that in a flat space you can describe points by choosing and origin point and then using vectors "based at" that point; but this is exactly what makes the situation so confusing.

Think for a moment about the unit sphere. The sphere itself is made up of points. At every point, there is also a space of tangent vectors, i.e. the set of all vectors based at that point which are tangent to the sphere at that point. These vectors represents all the different directions you travel away from their base point.

Now, just because two tangent vectors at different points on the sphere happen to be parallel and have the same magnitude in the ambient 3D space does not mean we should consider them "the same". Imagine a great circle between these two points, and that we have two vectors pointing along the great circle. It seems reasonable to say that these vectors are pointing in the "same direction", yet for most pairs of points they will not be parallel in the ambient 3D space.

The point is that we cannot compare vectors in different tangent spaces without further information or assumptions. It is clarifying to bring this attitude over to flat space.

So in exactly the same way, in flat 2D space we have points and we also have tangent spaces of vectors based at those point. Physically, you can think of points as representing positions of a particle. Tangent vectors represent thing like velocity, momentum, or acceleration of such a particle; or a force applied to a particle; or the value of a vector field (like electromagnetism) at that position.

We can represent points with, say, polar coordinates $(\rho,\varphi)$. We can also endow each tangent space with basis vectors however we wish; but at (almost) every point, notice how there is a unique line of constant $\rho$ and a unique line of constant $\phi$ passing through that point. So if we're using these coordinates, it seems natural to use bases for the tangent space which align with these lines. This is the idea behind the holonomic basis $e_\rho,e_\varphi$ associated with the coordinates $(\rho,\varphi)$. This is a basis for each tangent space, so $e_\rho$ and $e_\phi$ are functions of points in space, or equivalently functions of the coordinates $(\rho,\varphi)$.

(The proper holonomic basis has $|e_\rho| = 1$ but $|e_\varphi| = \rho$; this is because moving one unit of $\rho$ for fixed $\varphi$ move you one unit of distance, but moving one unit of $\varphi$ for fixed $\rho$ moves you a distance proportional to $\rho$. It is common to use the unit vectors $e_\rho,\hat e_\varphi$ in applications instead.)

An example: consider the physical situation of a ball tied to a string being spun around a particular point the string is fixed to. We choose this point as the origin. The the ball's position is described by $(\rho,\varphi)$ with constant $\rho$. It's velocity is always in the direction $\pm e_\varphi(\rho,\varphi)$. It's acceleration and similarly the force exerted by the string on the ball are in the direction $e_\rho(\rho,\varphi)$. If the string has some stretch to it, and the ball is moving in and out while it spins, then its velocity is some combination of $e_\rho$ and $e_\varphi$, something like $(\sin\varphi)e_\rho + v_0\hat e_\varphi$.

What $e_\rho$ and $e_\varphi$ do not describe is the position of the ball. Can they? Yes, but that is a pun specific to flat space which is best avoided. I would think of the fact of $\rho e_\rho(\rho,\varphi)$ being the position vector of the ball, when based at the origin, as merely a happy accident. You could also pick some other particular point $(\rho_0,\varphi_0)$ and the express the position　vector in terms of $e_\rho(\rho_0,\varphi_0)$ and $e_\varphi(\rho_0,\varphi_0)$; both are needed in this case and the resulting expression is more complicated.

Nicholas Todoroff
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  • Thank you very much - that is a pretty good explenation. Yes - come from graphics programming so talking about vectors as points is a mistake I often make. However I'm still not quite sure about my original question ... Often I see the polar vector (5;45°) written as $\vec c=5\vec e_x+45°\vec e_θ$ - how can that be true? I just don't see how adding those two vectors results in the targeted result? It should be something completely different, souldn't it? θ is not 45° for $\vec c$ then. – Robbe Oct 03 '24 at 19:22
  • @Robbe Yes, what you wrote is absolutely incorrect. Where have you seen this? – Nicholas Todoroff Oct 04 '24 at 00:24