If u and v are vectors in three dimensional space, then u × v is a three dimensional vector. Unlike the other operations, its operands and its result must be 3D vectors. The operation is not defined for 2D.
Definition of the Cross product u × v
The result is a vector perpendicular to the two operand vectors. But there are two directions that are perpendicular to both operands. Which one does the cross product give you? That is determined by the right hand rule, which is explained on the next page.
A wood screw is positioned perpendicular to a plank of wood. You wish drive the screw into the wood with a screw driver. Which direction must you twist the screw driver?