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Hi. The dot product is an operation that provides relations between vectors. This is axiomatically needed to close definition of linear space, but its major notion is to be able to compare 2 vectors. Else, how can we compare vectors? Comparison is made by dot product where we define metrics. The dot product is necessary to be able to "differentiate" vectors. As it was pointed out it also defines projections, but it is secondary. The first meaning is to define "magnitude" (or scalar quantity) to each vector (by projecting it onto itself).
The cross product is a mathematical notation to reflect vector mapping. What I mean is that unlike the dot product that maps 2 or more vectors in a scalar, the cross product maps vectors in vectors and it is inhere to simply describe what happens in nature. Else, we shall not be able togive full account to effects like momentum for instance or many force relations in mechanics.
Generally speaking cross product is not necessary for linear spaces and is an useful addition to it.
For linear spaces to be closed it is necessary that the linear combination of any vectors remains in there, yet any linear combination of 2 vectors defines and retains the plane. The cross product (besides being vector) is orthogonal to the plane of its parents. Like when you tighten the bolts of your car's tire you rotate in one plane (the gauge) and it translates to linear force upward.