Let's say a ball moves in an azimuthal path around a rotating disk. The Coriolis force on the ball is given by $-2m \omega \times v$. My intuition tells me that there would be no Coriolis force because the velocity is in the counter clockwise direction as is the direction of rotation. However, when I look at the cross-product in the Coriolis Force, the angular velocity points up given by the right-hand rule and the velocity points in the theta hat direction which is in the $xy$ plane so the cross-product would not equal zero. The only way the cross product could equal zero is if the velocity is also pointed straight up but the velocity can never point up. Where is the flaw in my train of thought?