It actually depends on the situation you want to use the multiplication. For some applications, dot product is used, while for others cross product is useful.
The dot product gives a scalar number, while a cross product gives us a vector only.
For example, to find the angle between two vectors, you can use cross product easily to find: cos = / |A||B|
While to find the vector perpendicular to the plane of the 2 vectors, or to calculate torque,it is given by its cross product only.