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So I have a query concerning position vectors and cylindrical coordinates.

In my electromagnetism text(undergrad) there's the following statements for

position vectors in cylindrical coordinates:

$$\vec r = \rho \cos\phi \hat x + \rho \sin\phi \hat y+z\hat z$$

I understand this statement, it's the following, I don't understand how a 3D position can be expressed thusly:

$$\vec r = \rho \hat \rho + z \hat z$$

Thanks for any insight and help!

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    $\begingroup$ Minor markup detail: cos\theta typesets as the product of four symbols ($cos\theta$) while \cos\theta typesets as the cosine of theta ($\cos\theta$). Similarly for \ln, \log and so on. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Mar 23, 2019 at 21:24

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Because

$$\hat{\rho}=\cos{\phi}\hat{x}+\sin{\phi}\hat{y}.$$

This is a unit vector in the outward (away from the $z$-axis) direction. Unlike $\hat{z}$, it depends on your azimuthal angle.

The position vector has no component in the tangential $\hat{\phi}$ direction. In cylindrical coordinates, you just go “outward” and then “up or down” to get from the origin to an arbitrary point.

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