Im trying to understand helmholts decomposition, and in order to do so, I feel the need to understand the different ways to apply the del operator to a vector valued function. The dot product and the cross product between two ordinary vectors are easy to understand, thinking about them as a projection onto another vector, or the area of the parallelogram between the vectors as magnitude with the resulting vector perpendicular to both original vectors. However, when we use the del vector, which is a vector of operators, this way of thinking gets harder. Is there a way of understanding these vector calculus operations by knowing about the geometrical properties of the dot and cross product, which in turn lends better understanding to the second order derivation operations on a vector functions? I know what curl and div is, that's not the problem, Im very interested in an analogy between whats happening with real vectors that's multiplied in these ways vs when the del operator is multiplied in these ways with a vector function. Thanks for any answer!!