Suppose we have a particle with orbital angular momentum $\vec L$ and spin $\vec S$ with quantum numbers $l=1$ and $s=1/2$ respectively. We know that the total angular momentum must have quantum number $j=1/2$ or $j=3/2$. This is being explained as "vector addition", so I've been trying to figure out if they really work like vectors, except they can only be in such directions so that their projection is quantized.
From now on we will consider the projection on the $z$ axis and let the projection of $\vec L$ be $1$ and the projection of $\vec S$ be $1/2$ ($\hbar$ is set to 1). {$L^2,S^2,L_z,S_z$} all commute, so we can do this.
By the law of cosine we can calculate the squared module of $\vec L +\vec S$, which is nothing else but $\vec J ^2$ and we know that it has to be either $3/4=0.75$ or $15/4=3.75$. Now the cosine law gives us only one solution (one angle and its opposite) if we want $\vec J ^2$ to be $15/4$ (and 0 solutions for $3/4$).
Does that mean that, after we observe {$L^2,S^2,L_z,S_z$}, the two momenta $\vec L$ and $\vec S$ not only have a particular angle with respect to the $z$ axis, but also a particular fixed angle with respect to each other (eventually swapped)?
Edit: there was a mistake.
