Why is vector division not possible?

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 #ox# as complex number 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]#

#=[a_1a_2-b_1b_2-c_1c_2-d_1d_2,#
#color(white)(0000) a_1b_2+b_1a_2+c_1d_2-d_1c_2,#
#color(white)(0000) a_1c_2-b_1d_2+c_1a_2+d_1b_2,#
#color(white)(0000)a_1d_2+b_1c_2-c_1b_2+d_1a_2]#

If this looks a bit like the expansion of matrix multiplication it is no coincidence. Quaternions can be represented by corresponding #4xx4# real matrices of the form:

#((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: