Why does multiplication of vectors not have a corresponding division?
It can have...
The normal kinds of multiplication of vectors that you have probably encountered are dot product and cross product.
Dot product does not support division since it makes two vectors into one scalar.
Cross product does not support division, since the cross product of two non-zero vectors in the same direction is zero.
However, they can be combined into quaternion multiplication in four dimensional space, creating a form of multiplication that does support division by any non-zero element.
See https://socratic.org/s/aCwSAEhJ for some more details.