Write down the commutation relations satisfied by the cartesian components of the total angular momentum operator $\mathbf{J}$.

In quantum mechanics an operator $\mathbf{V}$ is said to be a vector operator if, under rotations, its components transform in the same way as those of the coordinate operator r. Show from first principles that this implies that its cartesian components satisfy the commutation relations

$$\left[J_{j}, V_{k}\right]=i \epsilon_{j k l} V_{l}$$

Hence calculate the total angular momentum of the nonvanishing states $V_{j}|0\rangle$, where $|0\rangle$ is the vacuum state.