The centrifugal force vanishes if and only if the position vector $x$ is parallel to the angular velocity vector $\omega(x \| \omega)$. This means the particle lies directly along the axis of rotation. In this case, the condition for vanishing force, $\omega(\omega \cdot x)=\omega^2 x$, is satisfied, and the term $\omega \times x=0$ in the original formula.

<aside> ❓

1. Under what geometric condition does the centrifugal force on a particle of mass vanish in a rotating frame? </aside>