Vector addition is defined algebraically as the sum of vector components, and it's usually taught geometrically by drawing two little arrows, placing them head to tail, with the second arrow's tail at the first arrow's head, and drawing a third arrow from the tail of the first one to the head of the second one, and calling that third arrow the sum. That all works well in Cartesian coordinates, where the components encode translations, and the algebraic and geometric definitions coincide, but fails completely in polar coordinates, where the components encode a scaling and a rotation, and the algebraic and geometric definitions diverge.

In Cartesian to polar transformations, the unit vectors $\hat x$, $\hat y$ are transformed to $\hat r$, $\hat \theta$, with

$$\hat r = \cos\theta\hat x + \sin\theta\hat y$$

$$\hat \theta = -\sin\theta\hat x + \cos\theta\hat y$$

On diagrams, $\hat r$ is shown being normal to a circle and $\hat \theta$ tangent.

What is the meaning of $\hat \theta$? Is it a unit rotation (rotation by one radian) or is it a displacement by one radian in the tangent along the circle? Why is it usually drawn as a tangent instead of as a curve along the circle?

  • $\begingroup$ It's the unit vector tangent to the circle, representing the direction perpendicular to the radial direction. (The "hat" or caret generally indicates a unit vector.) It is just used as a basis vector, so it does not of itself indicate a "rotation by one radian" or a "displacement by one radian", any more than $ \ \hat{x} \ $ represents a displacement of one meter in the positive $ \ x-$direction. $\endgroup$
    – boojum
    May 13, 2022 at 20:43
  • $\begingroup$ $\hat{\theta}$ points in the direction you'd move if you increased $\theta$. $\endgroup$ May 13, 2022 at 20:47
  • $\begingroup$ I think it's a somewhat confusing notation. $\hat r,\hat\theta$ is just the rotation of $\hat x,\hat y$ by $\theta$. $\endgroup$
    – Berci
    May 13, 2022 at 20:55
  • 1
    $\begingroup$ The transformation $\hat \theta = -\sin\theta\hat x + \cos\theta\hat y$ is useful for certain applications, not for others. Perhaps you will find some clarification in my answer to a similar question. $\endgroup$
    – David K
    May 14, 2022 at 0:16
  • $\begingroup$ That is a very good answer, indeed. I can see now how the unit vectors $\hat r$ and $\hat \theta$ create a local Cartesian coordinate system in which the usual rules can be applied for reasoning about differentials. But in that answer you mention that you can't take a difference of two polar coordinates component-wise, and I don't get that, as I believe that works. $\endgroup$ May 14, 2022 at 0:49


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