Carpentry of Vectors

The cross product of two vectors is perpendicular to both; the right hand rule picks the one out of two possible perpendicular directions.

The diagrams show the right hand rule. The order of operands is important. The result of u × v points in the same direction a screw must point if twisting from u to v drives it into the wood.

Or, think of the direction your thumb points when your fingers curl around from u to v.