Why vectors cannot be divided?

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Vectors can be divided by scalars, but the division between vectors is AFAIK not defined. I would say it is because although there is only one logical way to define division between scalars, this is not the case with vectors.

This happens as well for multiplication, where dot product and cross product are two entirely different operations.

Ask yourself, how would you define vector division, and why? Would that be the only way? You could, for example, define such a thing as a "dot" division, as

a⃗ ÷b⃗ =i=1i=Laibia→÷b→=∑i=1i=Laibi

given a⃗ a→ and b⃗ b→ have a length of LL elements. But as you may consider, this would be quite arbitrary, and of no obvious usefulness.therefore its not divisible

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