LS

[SPelonis]

J=2 -> Parallel and pointing in the same direction
J=0 -> Parallel and pointing in the opposite direction
J=1 -> Perpendicular

L,S are vectors and we can easily add them, if they are parallel. This means that the first the cases are easily understood. For instance, let’s look at case J=2.

\vec{J} = \vec{L} + \vec{S} = 2a\hat{x} if they are pointing in the x direction.

If they are perpendicular (e.g. one in the x and another in the y direction) then:

\vec{J} = \vec{L} + \vec{S} = a(\hat{x} + \hat{y})
with a norm equal to a

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