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\vec v \cdot \vec u = v_1 u_2 + v_2 u_2 [/tex]? why not [tex]

\vec v \cdot \vec u = 3v_1 u_2 + 2v_1 u_2 [/tex]? I do see how defining the dot product as [tex]\vec v \cdot \vec u = v_1 u_2 + v_2 u_2 [/tex] helps solve problems in physics, but is that the only reason why the dot product is defined that way?

I have pretty much the same question about the cross product, i see how the definition of the cross product helps solve physics problems, but it still seems just made up to me. Is this the case or is there some other deeper mathematical meaning behind the cross and dot product that eludes me?

Also why are they called products? What makes these two operations multiplication?