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?
