Why it is not possible to have divisions of vectors?

[wasi-uz-zaman]

hi,please answer some question about vectors?
a) why it is not possible to have divisions of vectors?
b) is vector product of two polar vector is always axial vector?
c)what is vector product of one axial vector and polar vector?
thanks
wasi-uz-zaman

 

[Tenshou]

If you believe the "geometric" intuition of vectors then it is kinda obvious why, not? but if you dive a little deeper into the construction of a vector all it is a function which must obey these rules
$+ : E \times E \to E$
$+(x_1,x_2) \mapsto x_1+x_2$ and the axiom of a group with only this operation
[1]associative law of addition
[2]commutative law of addition
[3] inverse.
Also, you can only multiply scalars to vectors (cross product is a completely different operation which falls into something called the determinant function. this function has an inverse, if that is what you are asking.)
Could you clarify what you mean why axial vector and polar vector? as in rectangular and polar coordinates?

Edit: I mean that if you understand that Vectors are only additive groups then it should be helpful to understand that they aren't multiplicative groups.

 

[Viracocha]

a) a vector's not a number like we think of numbers, so you can't divide with them (although there are plenty of other operations in higher-level math)

don't know about b or c though...

 

[SteamKing]

It's not clear what the OP means by an 'axial' and a 'polar' vector.