Besides the angular momentum in each direction, we also have the total angular momentum:

$L^_{2}=L^_{x}+L^_{y}+L^_{z}$

Then you have to understand what each one of those does to the each atomic orbital:

- total angular momentum: determined by the azimuthal quantum number
- angular momentum in one direction ($z$ by convention): determined by the magnetic quantum number

There is an uncertainty principle between the x, y and z angular momentums, we can only measure one of them with certainty at a time. Video 71. ""Quantum Mechanics 7a - Angular Momentum I" by ViaScience (2013)" justifies this intuitively by mentioning that this is analogous to precession: if you try to measure electrons e.g. with the Zeeman effect the precess on the other directions which you end up modifing.

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