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In quantum mechanics a measurement changes the state. Now, that you find it contradictory that we may find a state to have spin aligned with the z-axis if we determined it to have spin aligned with the x-axis before suggests you are still thinking classically. In fact, if one examines the eigenstates, one finds that the states that are eigenstates of one of the components of angular momentum are completely scrambled in the other two components- the $x_+$-state has equal probability to have "up" or "down" for $S_y$ and $S_z$. You can't expect that an arbitrary eigenstate of $S_x$ is also one of $S_y$ or $S_z$ - the operators $S_x,S_y,S_z$ do not commute and therefore cannot share an eigenbasis (though they may share some eigenvactors). If we know that a vector is pointing along the $|x \rangle$ direction, we know for sure that it has zero component along the $|y \rangle$ direction. We can decompose them into $x$ and $y$ components call the $x$ and $y$ unit vectors $|x \rangle$ and $|y \rangle$. There's an intuitive way to see this, but to do it, we have to temporarily leave quantum mechanics and do some high school geometry.Ĭonsider vectors in the plane. You've found the sharpest reason why: a spin with definite $z$ component doesn't have a definite $x$ component. The correspondence between the two is subtle and nonintuitive generally, it's a bad idea to try to think of quantum spin as vectorial at all. Spin is not described by a three-component vector, it's described by a two-component spinor.