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## Main Question or Discussion Point

*why is vector division not possible?*

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- Thread starter Atizaz Ahmed
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Dale

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What is vector division?

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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

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It is possible. https://en.wikipedia.org/wiki/Geometric_algebrawhy is vector division not possible?

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Physics Footnotes

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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.

\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

Also keep in mind that whilst it is often

Last edited:

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Stephen Tashi

Science Advisor

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We must distinguish between "is not possible" and "is not defined". For example, in the standard definition of a "vector space", the division of vectorswhy is vector division not possible?

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.

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