Velocity in polar coordinates (again)

[tiago23]

Hey people, this question was already asked here [https://www.physicsforums.com/threads/velocity-in-plane-polar-coordinates.795749/], but I just couldn't understand the answer given, so I was wondering if some of you could help me by explaining it again. I don't really get Equation (or approximation) 1.11.3, how could the magnitude of Δer be equal (or similar) to Δθ.

 

[Chestermiller]

The easiest way to show this is to first resolve $e_r$ and $e_{\theta}$ into components in the cartesian coordinate directions:
$$e_r=e_x\cos{\theta}+e_y\sin{\theta}$$
$$e_{\theta}=-e_x\sin{\theta}+e_y\cos{\theta}$$
Are you OK with this so far?

 

[tiago23]

Hey @Chestermiller sorry it took so long for me to reply, internet access here is a bit precarious. I understand the derivation of this relation first decomposing the vectors into its components, and then differentiating it with relation to time, but the approach taken in the book (and the answer to the previous question) is the one I can't wrap my mind around. Sorry if this a bit capricious or demanding, I just want to understand the approach taken. :)