I was strolling through the neighborhood physics department recently, and I came upon an admonition on a bulletin board. Do not divide vectors the posting said, somewhat sternly. If A and B are vectors, than A/B is undefined.
Unfortunately this is not quite correct. To be fair, the context of the posting is the algebra one encounters in undergraduate physics courses. In the vanilla vector spaces in introductory mechanics or in a first course in linear algebra, e.g., vector division is verboten. But math is a mature and a rich field, and many very smart people have been investigating vector spaces and abstract algebras for more than a century. Some humility is always in order when thinking about absolute statements in algebra.
In a vector space over the field of real numbers R, it is possible to define an algebra in which a given vector has a multiplicative inverse. It's called geometric algebra, which is a kind of Clifford algebra. Dividing one vector by another is a resultant quantity called a quaternion. Quaternions were invented by William Rowan Hamilton in 1843. There's an introduction to geometric algebra book one can download for free. There is at least one physics book that uses geometric algebra, so this is not just mathematical abstraction.
The axioms of geometric algebra are quite simple:
These axioms lead directly to new quantities, imaginary unit vectors that are analogous to the square root of -1 that is introduced when extending the real numbers to complex numbers. So, to summarize, if we're strictly limited to a vector space in R3, then yes, vector division is not allowed. But it is possible to extend your vector space and introduce new quantities that do allow vector division, where dividing by a vector is defined as multiplying by its inverse. For further details, see this page in the book by Hestenes.