# Linear Algebra

## Main Question or Discussion Point

why is vector division not possible?

Dale
Mentor
What is vector division?

Physics Footnotes
Gold Member
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.

Last edited:
Stephen Tashi