What is vector division?

As already mentioned in the comments you have two ways of "multiplying" vectors. You have the dot product and the cross product. However, the dot product isn't a product.

When you multiply two rational numbers, you get a rational number. When you multiply two matrices, you get a matrix. When you multiply two complex numbers, you get a complex number. So you would want your product to satisfy that the multiplication of two vectors gives a new vector. However, the dot product of two vectors gives a scalar (a number) and not a vector.

But you do have the cross product. The cross product of two (3 dimensional) vectors is indeed a new vector. So you actually have a product. It is still a bit of a strange product in that it is not commutative. $\vec{x}\times\vec{y}$ isn't (always) the same as $\vec{y}\times\vec{x}$.

Now about division. If you have two real numbers $x$ and $y\neq 0$, we say that $\frac{x}{y} = z$ exactly when $x = yz$. So in that sense you could define a type of division of vectors.

However, again there are some problems with vectors. When we divide by a real number $y$, we can also consider this as multiplying by the inverse of $y$, that is, $y^{-1}$. The inverse of $y$ is that unique number $y^{-1}$ such that $yy^{-1} = 1$. The number $1$ is that "special" number that satisfies that $1x = x$ for all real numbers $x$. And you see that any (nonzero) number divided is $1$. The question is: what would the equivalent of $1$ be for vectors?

With vectors, you don't have such a "unit". There is no vector $\vec{1}$ such that the cross product of $\vec{1}$ with any other vector $\vec{x}$ is $\vec{x}$, that is, $\vec{1}\times \vec{x} = \vec{x}$.

So that is way we don't really have a division of vectors that "works" just like division of real numbers do.

You can define division of vectors, but as multiplication and division are related operations, you can only do so by choosing a definition of multiplication that allows it.

As has been pointed out, in vector algebra we typically just define the dot and cross products. For two vectors $a$ and $b$, the dot product $a\cdot b$ tells us how much the two vectors are parallel. The cross product $a \times b$ tells us how perpendicular the vectors are, and moreover, it tells us something about their relative orientation--about the plane that the two vectors lie in. I submit to you without proof that the dot and cross products contain all the relevant information possible from two vectors. In other words, if one knows $a$ and $a \cdot b$ and $a \times b$, then one can reconstruct $b$.

Indeed, the formula for doing so is something like

$$b = (a \cdot b) a/|a|^2 + (a \times b) \times a/|a|^2$$

It should be intuitive that $a/|a|^2$ somehow "undoes" these two products. If there were a candidate for $a^{-1}$, then $a/|a|^2$ would be it.

But how can we go about this in a formalized way? The answer is to define a new product, one that combines the properties of the dot and cross products into a single operation. This operation is called the geometric product.

Let $e_1, e_2, \ldots, e_n$ be an orthonormal basis for $\mathbb R^n$. The geometric product of vectors is defined as follows:

$$e_i e_j = \begin{cases} 1, & i = j \\ -e_j e_i, & i \neq j\end{cases}$$

When two basis vectors are the same and multiplied via the geometric product, the result is a scalar, and so we capture the behavior of the dot product. When the two basis vectors are orthogonal, the result is antisymmetric, and we capture the behavior of the cross product. It's important to note, though, that this antisymmetric part does not result in a vector--rather, it results in a new object we call a bivector. Think of it as an oriented planar subspace, just as vectors are oriented line-like subspaces through $\mathbb R^n$.

The geometric product is linear on its arguments, so we can find the geometric product of $a$ and $b$ just by breaking them down into components. Furthermore, the geometric product is associative, so we can find $ab$ and then multiply (on either the left or right) by another vector $c$, and so on. For now, though, we can restrict ourselves to the case of two vectors. The geometric product is often written as

$$ab = a \cdot b + a \wedge b$$

This wedge neatly avoids one problem with the cross product--it doesn't exist in dimensions outside of 3 or 7. The wedge (which produces the bivector part mentioned earlier) does exist in any number of dimensions, though.

Now then, the geometric product admits multiplicative inverses (essentially, division). See that $aa^{-1} = 1 \implies a^{-1} = a/a^2$, just as I observed before. Because the geometric product is associative, it is meaningful to say that

$$a^{-1} a b = (a^{-1} a) b = \frac{1}{a^2}aab = \frac{a^2}{a^2} b = b$$

where on the other hand, associativity gives us the freedom to group the products differently, like so:

$$b=a^{-1} a b = a^{-1} (ab) = a^{-1} (a \cdot b + a \wedge b) = \frac{1}{a^2} [a(a \cdot b) + a \cdot (a \wedge b)]$$

which is just the geometric algebra form of the decomposition I wrote earlier. Here, it follows just from the freedom to group products as one sees fit. This is a powerful technique in geometric algebra, useful for proving many identities (even up to vector calculus and beyond).

All that aside, what is the product of two vectors then, under the geometric product? It is a scalar and a bivector, as we've established. One name for the set of such objects is spinors. Spinors are useful for representing rotations, and indeed, the product $ab$ gives us a spinor corresponding to a rotation from $b$ into the direction of $a$. In 2 dimensions, such spinors have only two components, and these correspond to complex numbers. In 3 dimensions, such spinors have 4 components (1 scalar, 3 bivector--for the 3 planes of 3d space), and these spinors correspond to quaternions and so on. Thus, the geometric product gives great insight into the nature of rotations and how they can be built from vectors.

Given any vector $b$, you can find some nonzero $c,d$ with $b \cdot c=0$ and $b \times d=0$ (just take $c$ perpendicular to $b$, and $d$ parallel).

To say that $x=a / b$, where $/$ is a division operation that corresponds to dot product, should be equivalent to saying that $a = b \cdot x$. But if this is true, then also $a = b \cdot (x+c)$ where $b \cdot c=0$, so we should also say that $x+c=a/b$. That is, "dot-product division" is never uniquely defined, no matter the choice of $a$ and $b$. So it's not really a useful concept. Similar remarks apply to "cross-product division" — just replace $c$ by $d$.

On the other hand, there is (sort of) a definition of vector division based on scalar multiplication: if $a$ and $b$ are parallel vectors, then you can divide $a$ by $b$ to get a real number. Of course, this isn't defined for general pairs of vectors. But it is unique whenever it exists, which means it's occasionally a useful concept...