The Coriolis force is given by the vector equation $-2m \Omega \times v'$.
Where $v'$ is the velocity with respect to the rotating frame.
For the case of a projectile on the Earth, $\Omega$ is along the axis of rotation. It seems from this definition of the Coriolis force that a projectile fired in a northwards direction will have a $v'$ parallel to $\Omega$, so the right-hand rule would give a force of zero magnitude. Likewise, for a projectile fired in an eastwards direction, the right-hand rule would give a force directed towards the center of the Earth.
However, I know that the true result is that the northward projectile experiences an eastward force and the eastward projectile experiences a southward force. How is this compatible with the right-hand rule?
(This is NOT a duplicate of Is there an intuitive explanation for the Southward force caused by the Coriolis Effect on rotating spheres?, because my question is specifically about the vector expression of the force, which is not addressed in that post.)