Knowing dot and cross products can simplify the calculations for a lot of problems. They are useful only when vector are used.
The dot product is the projection of one vector onto another, resulting in a scalar.
The cross product is the product of two vectors, resulting in the production of a third, novel vector.
You might need to use it in EM and Kinetics, because it simplifies force-velocity calculations and allows you to skip messing around with all the various angles and cosines.