Short version :
How do you graphically represent a vector(without converting to Cartesian) given components in direction of $\hat r$ and $\hat \theta$ (unit vectors in polar coordinates)? and what is the correct way to calculate its magnitude? and why? (graphical explanation would be highly appreciated)
For example $\vec r = 10\hat r + 30\hat \theta$ .
Long version :
If we -for example- want to represent the vector $\vec r = 3\hat i + 2\hat j$ in Cartesian coordinates we can do something like this :
Simple vector representation in Cartesian plane (image).
As shown in the image we can imagine adding the $\hat i$ vectors one at a time and then adding the $\hat j$ vectors as these vectors have constant magnitude and direction relative to the axis.
With polar coordinates however, the first thing that comes to mind when representing vectors is the polar representation which uses a value for the magnitude of the vector ($r$) and a value for the angle which the vector makes with the polar axis ($\theta$) like so : Polar representation (image).
Based on that I thought that when representing a vector like this : $\vec r = 10\hat r + 30\hat \theta$ I would just use the value of the coefficients of the unit vectors as the value for $r$ and $\theta$.
According to these links (some might be a little off-topic), this is not the case :
- relationship of polar unit vectors to rectangular
- Norm of vector in cylindrical coordinates
- https://www.physicsforums.com/threads/magnitude-of-a-vector-in-polar-coordinates.884351/
- https://www.physicsforums.com/threads/describing-a-position-vector-with-polar-coordinates.883559/
And also it is stated that unit vectors in polar coordinates depend on the value of $\theta$
$$\hat{r} = cos\,(\theta)\,\hat{i} + sin\,(\theta)\,\hat{j}\ , \hat{\theta} = -sin\,(\theta)\,\hat{i} + cos\,(\theta)\,\hat{j}$$
Doesn't this mean that you can't represent the given vector ? as the unit vectors could be anything when not given $\theta$.
So is the representation going to look something like these (images below), or something different? :
- Identical to Cartesian
- 2 constant perpendicular unit vectors with different orientation than the axis
- Something complex (for example each unit vector pair has a different orientation)
And finally how is the magnitude calculated ? (as it is not the component in the direction of $\hat r$ according to the posts above)
Best Answer
$\hat{\theta}$ tells you the angle you need to be at from the positive $x$-axis, and $\hat{r}$ tells you how far you need to walk out from the origin. The magnitude is most certainly given by the $\hat{r}$ component. Perhaps your issue is that you're not adding/multiplying vectors in polar form properly. You cannot simply take, for example, $v_1 = 1\hat{r} + \pi\hat{\theta}$, and conclude that $v_1 + v_1 = 2\hat{r} + 2\pi\hat{\theta}$. Notice that the angle has been changed, which shouldn't happen for two vectors that are colinear.
It is better to represent vectors as $re^{i\theta}$. From which in our example we have, $v_1 + v_1 = e^{i\pi} + e^{i\pi} = 2e^{i\pi}$, which has an $\hat{r}$ of 2, and $\hat{\theta}$ of still $\pi$.