Why is vector division not possible?
It can be...
Because vector multiplication is not generally arithmetic, but it can be.
A simple example in two dimensions would be if you treat vectors as Complex numbers and define a multiplication
#[a, b] ox [c, d] = [ac-bd, ac+bd]#
Then there is a corresponding division of vectors:
#[a, b] -: [c, d] = [a, b] ox [c/(c^2+d^2), -d/(c^2+d^2)]#
A more advanced example - useful in mechanics - is the quaternions. Hamilton's quaternions form a 4 dimensional vector space over the real numbers with a natural (though non-commutative) definition of multiplication that makes them into a division algebra, with a natural definition of division.
So treating four dimensional vectors as quaternions, we would define multiplication as:
#[a_1, b_1, c_1, d_1] ox [a_2, b_2, c_2, d_2]#
If this looks a bit like the expansion of matrix multiplication it is no coincidence. Quaternions can be represented by corresponding
#((a, -b, -c, -d), (b, a, -d, c), (c, d, a, -b), (d, -c, b, a))#
Then division is basically multiplication by the inverse matrix.
For a very interesting related talk see: