# What is the advantage of using a polar coordinate system with rotating unit vectors?

What is the advantage of using a polar coordinate system with rotating unit vectors? Kleppner's and Kolenkow's An Introduction to Mechanics states that base vectors $$\mathbf{ \hat{r}}$$ and $$\mathbf{\hat{\theta}}$$ have a variable direction, such that for a Cartesian coordinates system's base vectors $$\mathbf{ \hat{i}}$$ and $$\mathbf{ \hat{j}}$$ we have

$$\mathbf{\hat{r}} = \cos \theta\ \mathbf{\hat{i}} + \sin \theta\ \mathbf{\hat{j}}$$

$$\mathbf{\hat{\theta}} = -\sin \theta\ \mathbf{\hat{i}} + \cos \theta\ \mathbf{\hat{j}}$$

Now, isn't counter-productive to define a coordinate system in terms of another? Why, at least in this book, we choose to use such a dependent coordinate system, instead of using a polar coordinate system employing a radius and the angle that this one forms with a polar axis, therefore independent of another coordinate system?

EDIT: Let me clarify that I'm not asking about the advantages of the polar coordinate system over the Cartesian one, but about the advantages of a polar coordinate system defined on rotating base vectors $$\mathbf{ \hat{r}}$$ and $$\mathbf{\hat{\theta}}$$ over another polar coordinate system where we employ a base vector $$\mathbf{ \hat{r}}$$ and the angle (hence a scalar and not a vector) that this one forms with a polar axis.

• I think you may have misunderstood: the coordinate system described here is the plain polar coordinate system. The fact that the relationship between $(\mathbf{\hat{r}}, \mathbf{\hat{\theta}})$ and $(\mathbf{\hat{i}}, \mathbf{\hat{j}})$ is not constant is just a fact of life. – dmckee Jul 26 at 17:10
• $\theta$ is a shorthand for $\tan^{-1}(y/x)$. This transform describes a polar coordinate system with the polar axis parallel to $\hat{\mathbf{i}}$, but one could set the polar axis to be anywhere else by using $\tan^{-1}(y/x) + \theta_0$ for some constant $\theta_0$ instead. – eyeballfrog Jul 26 at 17:37
• Please check my edit. – torito verdejo Jul 26 at 17:49
• There is no such thing as a polar coordinate system with constant basis vectors. The nature of the coordinate system is that the basis vectors depend on the location at which you evaluate them. – dmckee Jul 26 at 17:57
• Thank you, @dmckee. After thinking about describing velocity and acceleration I've understood that a polar coordinate system without base vectors would be a headache when facing operations, derivation and integration among/on vectors. – torito verdejo Jul 26 at 18:04