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What is the need for different sets of unit vector for different locations (positions) in Polar Coordinate System ?

Just as the two fixed unit vectors in Cartesian Coordinate System, why cannot we have two imaginary unit vectors in radial and anti clockwise directions ; to capture the magnitude (distance from the origin), and the angle ?

I guess one inconsistency that will arise if there are only two unit vectors is: What would be the angle of the unit vector in the radial direction ?

Does the need for different sets of unit vectors at different locations arise to reconcile the above inconsistency ?

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Prasad B is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$ Commented Nov 20 at 15:43
  • $\begingroup$ Vectors must satisfy scalar multiplication and vector addiction. Can you see why those properties fail for the pair $(r,\theta)$ of radius and angle? $\endgroup$ Commented Nov 20 at 16:06
  • $\begingroup$ Hello @CyclotomicField, thank you for that nudge. Two vectors ($r_1$, $\theta_1$), ($r_2$, $\theta_1$), both along the same line when added has an angle that is twice the original -- which is clearly wrong. $\endgroup$ Commented Nov 20 at 17:11
  • $\begingroup$ @PrasadB and the scalar multiplication also doesn't work because of $-1(r,\theta)$ which will give us a negative radius. So it's not just the angle term that's problematic. The main assumption about of a vector space is that they are defined over a field which is a strong algebraic constraint. It will eliminate many examples you'll find in math and physics. For example mass can't be a vector because it's non-negative. Even for things like position you have to choose an origin before you can turn the pairs $(x,y)$ it into a vector space so there is some work to be done. $\endgroup$ Commented 2 days ago
  • $\begingroup$ Now you need to be careful to distinguish between points in the plane and (tangent) vectors. When physicists work with $\hat r,\hat\theta$ to write vectors at the given point $(r,\theta)$, they are interested in the vectors (e.g., velocity and acceleration), and are not changing the base point. $\endgroup$ Commented 2 days ago

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