What Is the Commutator [Le, Lf] in Terms of e, f, and L?

[Monalisa]

Homework Statement

Let e and f be unit vectors. L_e = eL is the definition of the component of angular momentum in direction e. Calculate the commutator [L_e,L_f ] in terms of e, f and L

Homework Equations

[A,B]=(AB-BA)

The Attempt at a Solution

we know that L=r x p, in classical mechanics, and in quantum physics we have the operators for angular momentum in cartesian coordinates for example, but in my problem I have just two direction, e and f, and I am obtaining as answering 0. How can I do this exercise ? thanks

 

[TSny]

Welcome to PF!

Since you know the commutation relations for the Cartesian components of L, it might be a good idea to write out e$\cdot$L and f$\cdot$L in terms of the Cartesian components of L.

 

[Sigurdsson]

Are the unit vectors $\hat{e}$ and $\hat{f}$ some arbitrary unit vectors in Cartesian space? In that case, it might be easy to start with something simple such as $\hat{e} = \hat{x}$ and $\hat{f} = \hat{y}$ and then moving into a more general case.

 

[Monalisa]

but, and about the z component for example, if I do this in two coordenates, the answer will be zero, maybe is zero the solution, I do not know

 

[BvU]

Why zero ? Follow Sigurdsson's sound advice and post your workings, please.