The dot product is a very general type of product for use on a variety of different types of vectors. It gives the product of the length of one vector times the length of the projection of the other vector onto it. This product is just a number, not a vector. We use it so much in all sorts of problems that I hardly know where to start in describing its uses. Let me give you just one example from a recent problem in a course. Say you want to figure out how much the free energy G of some chemicals changes when a reaction occurs that makes changes in a list of different components. Call that list of changing numbers of different types of molecules a vector. Then call the list of the chemical potentials of those different types another vector. The change in G is given by the dot product of those two vectors.
The cross product is a much more specialized tool used only for 3 dimensional vectors, giving another 3D vector. Itís ordinarily used only on vectors representing things in geometrical space, not for more generalized vectors. One common use is to determine the flow of energy in an electromagnetic field via the Poynting vector, a cross product of the magnetic and electric field vectors.
operator gives how much a vector field is pointing out or in to some region.
The delx operator tells how much some 3D vector field twists around some point. Since the maximal twist occurs around some axis, the direction of the delx tells what the direction of that axis is. Del.
operating on an electric field gives the distribution of the charged sources of that field. Delx operating on a magnetic field gives the distribution of current sources of that field- and those currents themselves have directions, unlike charges.
(published on 10/22/2007)