I am recently studying the two-state system in quantum mechanics.
As I learned, in the Hilbert space of a spinless particle, the relation between a scalar function and a ket state is satisfied as,
$$u(\vec r)=\langle\vec r|u \rangle$$
where $|r \rangle$ is the eigenstate of the position operator and $|u \rangle$ is the ket state of the spinless particle.
I wonder the similar relation can also be obtained with the two-state or multi-state ket. For example, if $|u \rangle$ is the two-state spin state as $|u \rangle = a|+ \rangle + b|- \rangle$, what is the result of $\langle\vec r|u \rangle$?
I think it should be ascalar as $a\langle\vec r|+ \rangle + b\langle\vec r|- \rangle$ because of the definition of the inner product, but I cannot find its physical meaning. Also, what is the corresponding scalar function of $\langle\vec r|+ \rangle$?