I was told that the direction of the cross product is an arbitary convention to give rotation a "direction" + for one direction and - for the other. That it is simply a book keeping device to make sure different rotation directions are given different signs. It seemed to be the case as I am able to do most of the rotational problems without using these mysterious "vectors out of the page". I merely used different signs for counterclockwise vs clockwise, which doesn't need the idea of vectors at all. counterclockwise vs clockwise as a model suffices completely. So I did the "bookkeeping" without any vectors. Basically, using rotational vectors in those 2d problems really are against the principle of Ocam's Razor. But with gyroscopic motion in 3d, you really do need the vectors to describe the motion. So the vectors are not an imagined construct. That they are somehow real? Is it even possible to understand gyroscopic motion without vectors?