I Vector Division: An Explainer of Why It's Not Possible

[Atizaz Ahmed]

why is vector division not possible?

 

[Dale]

What is vector division?

 

[Kiki]

Maybe matrix inversion can be considered vector division. However, only square matrices are invertible.

Here is some interesting content about so-called "matrix division" in Matlab, although I suspect that the term matrix division is not used in mathematics.
http://www.mathworks.com/help/matlab/matlab_prog/array-vs-matrix-operations.html

 

[micromass]

why is vector division not possible?

It is possible. https://en.wikipedia.org/wiki/Geometric_algebra

 

[Physics Footnotes]

For two-dimensional vectors you can define division easily by representing a vector $[a,b]$ by a complex number $a+bi$ and using complex division:$$
\begin{bmatrix}
a\\
b
\end{bmatrix}/
\begin{bmatrix}
c\\
d
\end{bmatrix} := (a+bi)(c+di)^{-1}
$$
For a more general solution, any vector space with an inner product can be embedded (in a geometrically natural way) inside a larger structure called a Clifford Algebra in which you can divide by a vector $v$ provided $v\cdot v \neq 0$ (which happens often in special relativity!). The mathematics of these larger systems is known as Geometric Algebra.

Also keep in mind that whilst it is often possible (and kinda fun!) to define non-standard mathematical operations, the real question you need to ask is whether it is useful to do so; especially if the context is physics. It turns out that geometric algebra is an extremely valuable tool in almost every branch of physics.

 

[Stephen Tashi]

why is vector division not possible?

We must distinguish between "is not possible" and "is not defined". For example, in the standard definition of a "vector space", the division of vectors is not defined . Some teachers may present this fact by saying that the division of vectors "is not possible" in order to advise their students not to waste their time trying to divide vectors when doing homework problems.

From a very advanced point of view there can be examples of things that are both "vector spaces" and also more complicated mathematical structures. If you are told about the more complicated structure then you may also be told about a way to do division in it.