Writing vectors in the $\hat{r}$ and $\hat{\theta}$ basis

Question

I am having a bit of confusion about writing down vectors in the $\hat{r}$, $\hat{\theta}$ basis. Let's say I want to write $-1$ $\hat{i}$ $+$ $1$ $\hat{j}$ in the $\hat{r}$, $\hat{\theta}$ basis. Could I write $1$ $\hat{r}$ $+$ $1$ $\hat{\theta}$ so long as I specify that $\hat{r}$ is "at" $\pi/2$ radians? (...and in the above question $\hat{\theta}$ would be "at" $\pi$ radians)

Could I have described the exact same point with $\sqrt2\hat{r}$ $+$ $0$ $\hat{\theta}$ so long as I specify this time that $\hat{r}$ is "at" $3\pi/4$ radians? (...and $\hat{\theta}$ would be "at" $5\pi/4$ radians, but we wouldn't move along the $\hat{\theta}$ direction this time).

Answer 1

Yes, you can do so.

Remember the vectors $\hat r,\hat\theta$ are defined by the equations $$\begin{cases}\hat r=\cos\theta\hat i+\sin\theta\hat j\\ \hat\theta=-\sin\theta\hat i+\cos\theta\hat j.\end{cases}\tag{1}$$

This is equivalent to $$\begin{cases}\hat i=\cos\theta\hat r-\sin\theta\hat\theta\\ \hat j=\sin\theta\hat r+\cos\theta\hat\theta.\end{cases}\tag{2}$$

To see why this is true, remember that each of $\hat r,\hat\theta$ are obtained by rotating each of $\hat i,\hat j$ by $\theta$ radian anti-clockwise (which is what $(1)$ does). So we can rotate each of $\hat r,\hat\theta$ by $-\theta$ radian anti-clockwise to obtain $\hat i,\hat j$ (which is what $(2)$ does).

So $-1\hat i+1\hat j=-(\cos\theta\hat r-\sin\theta\hat\theta)+(\sin\theta\hat r+\cos\theta\hat\theta)=(\sin\theta-\cos\theta)\hat r+(\sin\theta+\cos\theta)\hat\theta$. Plug in the value $\theta$ and you can express $-1\hat i+1\hat j$ in terms of $\hat r,\hat\theta$.

For case 1, you put $\theta=\frac{\pi}{2}$, giving $(\sin\theta-\cos\theta)\hat r+(\sin\theta+\cos\theta)\hat\theta=\hat r+\hat\theta$. For case 2, you put $\theta=\frac{3\pi}{4}$, giving $(\sin\theta-\cos\theta)\hat r+(\sin\theta+\cos\theta)\hat\theta=\sqrt2\hat r$.

Answer 2

You need a combination. You are correct in saying that your value of $\hat{r}$ needs to be $\sqrt{2}$, but by saying $0\hat{\theta}$ you are saying that your point lies in the $\hat{i}$ direction. Saying that it is "at" $3\pi /4$ radians is precisely what $3\pi /4 \hat{\theta}$ means.

Your answer in polar coordinates $(r, \theta)$ should be $\left(\sqrt2 , \frac{3\pi}{4}\right).$

Think of it in terms of $re^{i\theta} = r\cos\theta\hat{i} + r\sin\theta\hat{j}$

Answer 3

The question indeed originated in physics.stackexchange and the use of symbols here is very confusing. @edm considers $\hat{r}$, $\hat{\theta}$ and (i,j) as two cartesian coordinate systems where one is rotated by $\theta$ from the other. The symbols $\hat{x_1}$, $\hat{y_1}$, $\hat{x_2}$ and $\hat{y_2}$ can be applied just as well.

In physics, this coordinate system is usually related to a non-inertial system, for example a continuously rotating coordinate system following an object in circular motion. In that case, the object is always at r$\hat{r}$.

I usually refrain from using (r,$\theta$) in favor of (r,n) where n is a unit vector orthogonal to r. This clarifies the ambiguity between angle $\theta$ and vector $\theta$.

Answer 4

This is obviously a very old question but to consider $\hat{r}$ and $\hat{\theta}$ as a rotation transformation on $\hat{i}$ and $\hat{j}$ is terribly wrong. By your definition $\hat{r}$, $\hat{\theta}$, $\hat{i}$ and $\hat{j}$ all belong to the cartesian coordinate system, but you are giving the unit vector different names to refer to two Cartesian coordinate systems. If that’s what you are looking for, call them as $\hat{a}$ and $\hat{b}$ instead of $\hat{r}$ and $\hat{\theta}$.

You are wrongly handling the polar coordinates, which have the basis $\hat{r}$ and $\hat{\theta}$. If I were to write $-1\hat{i}+1\hat{j}$ in polar coordinate it indeed will be $\sqrt{2}\hat{r}+\frac{3\pi}{4}\hat{\theta}$. This is because fundamentally the polar coordinate interprets a vector through magnitude and angle. This thread does not debate this issue very well and as the person asking this question you have swiftly decided what is right or wrong. When you right $\hat{r}$ and $\hat{\theta}$, what are you referring to - basis vectors of a polar coordinate system or a new definition of basis vector as defined by the rotation matrix? It is also very wrong to say “it is a common misconception that $\hat{\theta}$ is intrinsically linked to angle.” $\hat{\theta}$ component refers to the angle of a vector whose magnitude is given by the magnitude of the $\hat{r}$ component. Both $\hat{r}$ and $\hat{\theta}$ are unit vectors.