# Position Vectors in Cylindrical Coordinates

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!

• 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. – dmckee --- ex-moderator kitten Mar 23 '19 at 21:24

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