Vector by Vector division I found quite some questions on Vector by Vector division and most of them focused on the fact that there are different Vector-Vector products and for the cross product the division would not make sense.
But what about Vector-Vector division as an inverse to Vector-Scalar multiplication?
What are the problems of defining the partial function $\div : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ such that
$$
  a \div b =
  \begin{cases}
    s &\text{ such that } s b = a \\
    undefined &\text{ if no such $s$ exists}
  \end{cases}
$$
And (assuming there were no problems with that definition), how unexpected would it be to find such a definition in the wild?

As some people have asked about the motivation for this:
The motivation comes from a C++ codebase, so not super mathy. We want to see if a Ray $\langle start, dir \rangle$ hits a Point $p$ and a simple implementation for this is to check if there exists a scalar $s$ such that $p = start + s \cdot dir$. While thinking about the name for the operation that finds this $s$, we thought about just using the division operator. So the question for us is "is $\div$ a good name for this operation?"
 A: Making a definition like this is fine. Calling it division is a bit unusual: if I were going to talk about this out loud, I'd call this a "$\mathbf b$-to-$\mathbf a$ scaling factor".
In mathematical writing, I wouldn't want to completely identify it with division, because it messes with established expectations. Vectors appear reasonably frequently in quotients: for example, take the Sherman-Morrison formula $$(A + \mathbf u \mathbf v^{\mathsf T})^{-1} = A^{-1} - \frac{A^{-1}\mathbf u \mathbf v^{\mathsf T}\!A^{-1}}{1 + \mathbf v^{\mathsf T}\!A^{-1}\mathbf u}.$$ What always happens in such cases is that the denominator simplifies to a scalar, and division by a scalar is fine.
With vector-by-vector division, there's a small chance of confusion about what operation is happening, but that's honestly unlikely. But once the denominator could be a vector, this takes away lots of context clues about what's going on, and makes expressions harder to interpret even if they're technically unambiguous.
The most likely way I'd talk about the expression you want to call $\mathbf a \div\mathbf b$ is as follows. Rather than say "We define the very important quantity $x = e^{\mathbf a \div\mathbf b}$, when this is defined" I would say "When $\mathbf a = \lambda \mathbf b$ for some $\lambda \in \mathbb R$, we define the very important quantity $x = e^{\lambda}$". Or whatever. Second best is writing something that makes it clear it's my own definition: writing it as $\textrm{div}(\mathbf a, \mathbf b)$, for example.

So what does that add up to for the C++ codebase? I don't recommend overloading the / operator, and personally I'd call it something like scalingFactor but many names could work.
