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Is there a way to correctly predict the direction of the unit radial vector and the unit transverse vector in problems like the one below or is it just better to take a guess and solve the problem base on your guess?

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The unit radial vector is always away from the origin. The unit tangential vector is always anti-clockwise around the origin such that $\hat{r}\times\hat{\theta}$ is out of the diagram. Not knowing what the problem is about, I'd probably do it in Cartesian and then convert to polar unless there is something inherently circular about the problem. So if you need a "guess," it would be better understood to be a definition of the coordinate system. Alternatively, you may need to consider a general angle $\theta$ to solve your problem.

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As a rule, $\hat{u}_X$ is always pointing in the direction along which $X$ grows. It works when $X$ is a linear parameter ($x$, $z$, $r$...) as well as when it's an angular parameter ($\theta$, $\phi$...)

So you can either decide the orientation of the parameter and deduce the orientation of the vector, or vice-versa.

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