Why is vector division not possible?

Question

Why is vector division not possible?

1 Answer

Impact of this question

6722 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-20T11:40:21

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 $\otimes$ $ox$ as complex number multiplication:

$[a, b] \otimes [c, d] = [a c - b d, a c + b d]$ $[a, b] ox [c, d] = [ac-bd, ac+bd]$

Then there is a corresponding division of vectors:

$[a, b] \div [c, d] = [a, b] \otimes [\frac{c}{c^{2} + d^{2}}, - \frac{d}{c^{2} + d^{2}}]$ $[a, b] -: [c, d] = [a, b] ox [c/(c^2+d^2), -d/(c^2+d^2)]$

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:

$[a_{1}, b_{1}, c_{1}, d_{1}] \otimes [a_{2}, b_{2}, c_{2}, d_{2}]$ $[a_1, b_1, c_1, d_1] ox [a_2, b_2, c_2, d_2]$

$= [a_{1} a_{2} - b_{1} b_{2} - c_{1} c_{2} - d_{1} d_{2},$ $=[a_1a_2-b_1b_2-c_1c_2-d_1d_2,$
$0000 a_{1} b_{2} + b_{1} a_{2} + c_{1} d_{2} - d_{1} c_{2},$ $color(white)(0000) a_1b_2+b_1a_2+c_1d_2-d_1c_2,$
$0000 a_{1} c_{2} - b_{1} d_{2} + c_{1} a_{2} + d_{1} b_{2},$ $color(white)(0000) a_1c_2-b_1d_2+c_1a_2+d_1b_2,$
$0000 a_{1} d_{2} + b_{1} c_{2} - c_{1} b_{2} + d_{1} a_{2}]$ $color(white)(0000)a_1d_2+b_1c_2-c_1b_2+d_1a_2]$

If this looks a bit like the expansion of matrix multiplication it is no coincidence. Quaternions can be represented by corresponding $4 \times 4$ $4xx4$ real matrices of the form:

$⎛ ⎜ ⎜ ⎜ ⎝ \begin{matrix} a & - b & - c & - d b & a & - d & c c & d & a & - b d & - c & b & a \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎠$ $((a, -b, -c, -d), (b, a, -d, c), (c, d, a, -b), (d, -c, b, a))$

Then division is basically multiplication by the inverse matrix.

For a very interesting related talk see:

Science

Math

Humanities

... and beyond

Why is vector division not possible?

1 Answer

Explanation:

Related questions

Impact of this question