Gfrktyl

It is given that L and Lz,Lx,Ly commute.(L is total angular momentum, Lx is angular momentum along x axis).
So, I can simultaneously know the value of let's say L and Lz. But, if I perform huge no of measurements and in a certain measurement, I get the value of L = Lz, then I know for certain that Lx and Ly are 0.
But, according to the uncertainty principle, I can't know the exact values of any two of Lx, Ly and Lz.
So, where did I go wrong?

You can't get

$$leftlVertvec LrightrVert = L_z $$

for non-zero $l$, since:

$$leftlVertvec LrightrVert = hbarsqrt{l(l+1)} $$

while the maximum value of $L_z$ is

$$ (L_z)_{mathrm{max}} = hbar l $$

Also: The maximal state is:

$$ Y_l^l(theta, phi) propto sin^l{theta}e^{ilphi} $$

which is not an eigenstate of $L_x$, nor $L_z$.

JEB's answer is correct: you can't do a huge number of measurements so that out of luck in one of them you will find

$$||vec L|| = L_z$$

You could try another way: since the reference frame is arbitrary, you could just choose it having the z-axis parallel to the angular momentum vector. This won't work either, because for doing so you would have to know where the angular momentum vector is pointing, and this would require simultaneous knowledge of its three components.

To add to the others,

L does not commute with $L_x$, $L^2$ does. Also, one is a scalar operator, the other is a vector.

Angular momentum is the product of position vector and momentum vector. So this scenario can serve as prototypical example of uncertainty principle arising in a single entity.

To elaborate on above, if you randomly pick a unit vector an infinite number of times, you'll never match the direction of the Angular momentum.