The difficulty is that there are more than one vector that when dotted with (a,b) yields 6, for example. We know this because the dot product of (a,b)·(c,d) is ac+bd=6. If we know a and b there are still two unknowns in the equation ac+bd=6. Any pair if numbers which satisfy this equation would qualify as the quotient of 6·/(a,b). Therefore dot division is not defined. Likewise there are infinitely many vectors which when crossed with (a,b,c) give us (d,e,f). Consider that AXB=C=|A||B|sin(q), where q is the angle from A to B. Any vector V in the plane of A and B where |V|sin(u)=|B|sin(q) will yield the same cross product. Again there are two unknowns, |V| and u, in the equation so there are infinitely many answers. Therefore cross division is also undefined.