Let $\vec{v}\in\mathbb{R}^2$ be a vector in cartesian coordinates that can be written as the linear combination of its vector components and the unit base vectors $(\vec{e}_1,\vec{e}_2)$, e.g.:
$$ \vec{v}=x\,\vec{e}_1+y\,\vec{e}_2 \,.$$
My understanding of coordinate transforms is still pretty limited, but I understand there are some important distinctions to be made:
- The vector itself is different from the vector components
- If we change to a different coordinate system, the vector components have to change
- If we change to a different coordinate system, the base vectors have to change as well
So, the linear combination in polar coordinates
$$ \vec{v}=r\,\vec{e}_r+\theta\,\vec{e}_\theta \,.$$
should yield the same vector as before. To see if that's true, I chose $\vec{v}=(x,y)^T=(1,1)^T$.
First, let's compute the unit base vectors $(\vec{e}_r,\vec{e}_\theta)^T$ in polar coordinates. We have
$$ \vec{v}=\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} r\,\cos\theta \\ r\,\sin\theta \end{pmatrix} \,.$$
With that, the unit base vectors in polar coordinates become
$$ \vec{e}_r = \frac{1}{h_r}\frac{\partial \vec{v}}{\partial r} = \begin{pmatrix}\cos\theta \\ \sin\theta \end{pmatrix}\quad , \quad \vec{e}_\theta = \frac{1}{h_\theta}\frac{\partial \vec{v}}{\partial \theta} = \frac{1}{r}\begin{pmatrix} -r\sin\theta \\ r\,\cos\theta \end{pmatrix} \,.$$
I did not want to "silently" apply the scale factors $h_r=1$, $h_\theta=r$ to make it more clear how to get from the base vectors to the unit base vectors.
The vector components in polar coordinates are
$$ \begin{pmatrix} r \\ \theta \end{pmatrix} = \begin{pmatrix} \sqrt{x^2+y^2} \\ \arctan{y/x} \end{pmatrix} = \begin{pmatrix} \sqrt{2} \\ \pi/4 \end{pmatrix} $$
Now, all that is left is to plug in all that we have and we should get the vector that we started with:
\begin{align}
\vec{v} = r\,\vec{e}_r + \theta\,\vec{e}_\theta &= \sqrt{2}\begin{pmatrix} \cos\pi/4 \\ \sin\pi/4 \end{pmatrix} + \frac{\pi}{4}\begin{pmatrix} -\sin\pi/4 \\ \cos\pi/4 \end{pmatrix}= \\[8pt]
&=\begin{pmatrix} 1 \\ 1 \end{pmatrix} + \frac{\pi}{4\sqrt{2}}\begin{pmatrix} -1 \\ 1 \end{pmatrix}
\end{align}
which is not the same vector that we began with in cartesian coordinates. What did I miss?
Best Answer
You are confusing coordinates of a point with components of a vector. It so happens that in flat space points can be represented with position vectors, but generally you need to keep vectors and points distinct. A point is a location and a vector is a direction from a given location. All of this to say that the expression $r\vec e_r + \theta\vec e_\theta$ makes no sense.
Remember that $\vec e_r$ and $\vec e_\theta$ depend on $r, \theta$; this is because they give a basis to specify directions (vectors) from the location (point) with coordinates $(r,\theta)$.
It so happens (in what could be considered a coincidence) that the position vector of a point with coordinates $(r,\theta)$ is $r\vec e_r(r,\theta)$.