Polar coordinate basis vectors

[Kimblee]

This was exactly what I was looking for. Cleared all my doubts.

[Toastbikini]

Well explained from the ground-up. Thanks a lot!

[brno]

thank you. I was confused as to finding the components of the angle unit vector. Your explanation using the dot product explained it perfectly

[pferrel]

I alway thought the basis for polar coordinate system was r=1, theta=1. Cartesian bases are [x=1, y=0], [x=0, y=1] right? I suppose we can have the bases of polar be anything that works but why use tan(theta)? This makes the basis in tan(theta) change as a function of theta? If the basis was theta=1 wouldn't this describe any vector without making the second basis a function of theta, it would be just one radian or degree or whatever, a number, not a function.

[markkennedy]

Hi, can you explain why the position vector can never be described as a linear combination of r hat and theta hat whereas the velocity and acceleration vectors derived from the position vector are described in terms of a linear combination of r hat and theta hat.

Indeed it seems velocity and acceleration vectors at each position are uniquely suited to this coordinate system since they are true vectors unlike the position vector (which starts at the origin and therefore only has an r hat component). This difference (between position vectors and their velocity/acceleration counterparts) seems to extend to the ability to take dot products in this coordinate system as well: dot products don't work for position vectors.

Can you shed light on what all this means. Is there a deeper physical significance associated with this difference in the treatment of vectors which doesn't happen for the Cartesian coordinate system. I heard reference to velocity and acceleration being true vectors in the tangent spaces of each point etc and this fits well with a changing basis at each point (as with this coordinate system). I hope you can shed light on this.