# Why is vector division not possible?

Jan 20, 2017

It can be...

#### Explanation:

Because vector multiplication is not generally arithmetic, but it can be.

A simple example in two dimensions would be if you treat vectors as Complex numbers and define a multiplication $\otimes$ as complex number multiplication:

$\left[a , b\right] \otimes \left[c , d\right] = \left[a c - b d , a c + b d\right]$

Then there is a corresponding division of vectors:

$\left[a , b\right] \div \left[c , d\right] = \left[a , b\right] \otimes \left[\frac{c}{{c}^{2} + {d}^{2}} , - \frac{d}{{c}^{2} + {d}^{2}}\right]$

A more advanced example - useful in mechanics - is the quaternions. Hamilton's quaternions form a 4 dimensional vector space over the real numbers with a natural (though non-commutative) definition of multiplication that makes them into a division algebra, with a natural definition of division.

So treating four dimensional vectors as quaternions, we would define multiplication as:

$\left[{a}_{1} , {b}_{1} , {c}_{1} , {d}_{1}\right] \otimes \left[{a}_{2} , {b}_{2} , {c}_{2} , {d}_{2}\right]$

=[a_1a_2-b_1b_2-c_1c_2-d_1d_2,
$\textcolor{w h i t e}{0000} {a}_{1} {b}_{2} + {b}_{1} {a}_{2} + {c}_{1} {d}_{2} - {d}_{1} {c}_{2} ,$
$\textcolor{w h i t e}{0000} {a}_{1} {c}_{2} - {b}_{1} {d}_{2} + {c}_{1} {a}_{2} + {d}_{1} {b}_{2} ,$
color(white)(0000)a_1d_2+b_1c_2-c_1b_2+d_1a_2]

If this looks a bit like the expansion of matrix multiplication it is no coincidence. Quaternions can be represented by corresponding $4 \times 4$ real matrices of the form:

$\left(\begin{matrix}a & - b & - c & - d \\ b & a & - d & c \\ c & d & a & - b \\ d & - c & b & a\end{matrix}\right)$

Then division is basically multiplication by the inverse matrix.

For a very interesting related talk see: