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In general, the coordinates of a vector are defined as the projections of it onto the coordinate axis. Moreover, in a polar coordinate system, the basis vectors $\hat e_\phi$, $\hat e_r$ depend on the location.

For concreteness, let's assume that we want to describe an object that starts at $\vec a=(1,1)_{x,y}=(\sqrt{2},45^\circ)_{r,\phi}$ and ends at $\vec a=(1,-1)_{x,y}=(\sqrt{2},-45^\circ)_{r,\phi}$.

enter image description here

How can we understand the coordinates in the polar coordinate system in terms of projections onto the coordiante axis, $\hat e_\phi$, $\hat e_r$? In particular, which basis vectors do we use? (This is nontrivial since the basis vectors $\hat e_\phi$, $\hat e_r$ depend on the location. Therefore, if we use the basis vectors at the location our vectors point to, we always find zero for the projection onto $\hat e_\phi$ which is obviously wrong.)

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What you've found is the difference between affine spaces and vector spaces. Affine spaces are just a bunch of points with no origin (though your coordinates may have a bunch of zeros, it doesn't mean anything physically, unlike the null vector in a vector space, with is the addition identity).

In affine spaces, vectors come in as the difference between points, so when using Cartesian coordinates we can substitute the point $(x, y)$ with the vector defined by:

$$ \vec v_{(x, y)} \equiv (x, y) - (0,0)$$

which then gives us the false impression that a point is a vector.

It is not.

When you treat it like a vector and generalize coordinates, you run into the very problem that generated your question.

It's not exactly clear what your are asking, but it looks like what is well known as "The First Geodetic Problem" (https://en.wikipedia.org/wiki/Geodesy#Geodetic_problems), which is:

"Given a point (in terms of its coordinates) and the direction (azimuth) and distance from that point to a second point, determine (the coordinates of) that second point."

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  • $\begingroup$ The coordinates of the two points indicated in the illustration above are $(\sqrt{2},45^\circ)_{r,\phi}$ and $(\sqrt{2},-45^\circ)_{r,\phi}$. But these aren't the projections of the vectors $\vec a$ and $\vec b$ onto the coordinate axes at the corresponding locations. As we can see from the figure, the projections yield $(\sqrt{2},0^\circ)_{r,\phi}$ and $(\sqrt{2},0^\circ)_{r,\phi}$. Thus my question really is, in what sense can we understand the coordinates $(\sqrt{2},45^\circ)_{r,\phi}$ and $(\sqrt{2},-45^\circ)_{r,\phi}$ in terms of projections onto coordinate axes? $\endgroup$
    – jak
    Commented Oct 13, 2019 at 15:05
  • $\begingroup$ Or, does this only make sense when we consider a vector field $\vec v ( \vec x)$ instead of an isolated vector? The vector field assigns a vector to each point in our, say, two dimensional real space $\mathbb R^2$? I can imagine that in this case, we need to use the projections onto the coordinate axes at the corresponding locations to determine the coordinates of $\vec v(\vec x)$ in spherical coordinates?! $\endgroup$
    – jak
    Commented Oct 13, 2019 at 15:10
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    $\begingroup$ @jak Points live in affine spaces, not vector spaces. From wikipedia's affine space entry: "in an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point." And visa versa. Points are not vectors and don't (can't) have projections onto other vectors. $\endgroup$
    – JEB
    Commented Oct 13, 2019 at 16:35

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