# Cartesian basis vectors in polar coordinates

I'm trying to solve a problem which includes a spinning object and I want to find it's velocity vector with respect to an object moving in a straight line on the the x axis with constant velocity.

I can represent the spinning object's velocity in cartesian coordinates and subtract from it the velocity of the moving object. But that is a method that doesn't scale if I have a more complex spinning shape. Is there a way to convert the object moving in a straight line to polar coordinates?

What is the x basis vector in polar vector form?

Edit: figured it out. The relation I was looking for was:

$$\hat{x}=cos(\theta)\hat{r}-sin(\theta)\hat{\theta}$$

• Are you familiar with the Jacobian matrix? The relation between the basis vectors in polar and Cartesian coordinates is closely related - you can construct it from partial derivatives of one set of variables with respect to the other - but the matrix must have determinant 1 (it's orthogonal). – Josh McK Dec 13 '18 at 17:45
• It's a lot less sophisticated. I figured it out and edited the post. – Yizhar Amir Dec 13 '18 at 18:11