Vector division

Question

I've been trying to understand how vectors work and this is my question so far: Is it possible to divide two parallel vectors? Is it possible to divide a vector by a scalar?

Answer 1

It is certainly possible to divide a vector by a nonzero scalar. In this case, you simply are dividing each component individually, e.g.:

$$(2,4,6)/2 = (1,2,3)$$

You can think about scalar division by $c$ as a scalar multiplication by $1/c$.

However, vector division by another vector is not usually well-defined. This is because cross products can not be uniquely inverted, in general.

Answer 2

We define the quotient of two numbers as the number that, when multiplied with the divisor, produces the dividend - provided such a number exists and is unique.

So nothing prevents you from transfering this definition to vectors. So the quotient of two vectors $\vec u$ and $\vec v$ should be a something $\xi$ such that $\vec u$ is $\xi\cdot \vec v$ (or perhaps $\vec v\cdot \xi$). What kind of object could $\xi$ be? Not a vector (unless in the special cas that you work in three dimensions and use the cross product as product). It could be a linear map. Or it could be a scalar. The linear map would not be unique (except in case of one dimension), so let's go with the scalar. That would be unique as long as $\vec v\ne\vec 0$ (sounds familiar), but exist only when $\vec u$ and $\vec v$ are linearly dependent (or parallel). So here you have your division of vectors.

For dividing a vector by a scalar, things are easier: We want $\frac{\vec u}{a}$ to be some object $\xi$ with $\vec u=a\cdot \xi$ (or perhaps $\xi\cdot a$). The obvious choice is to let $\xi$ be the vector $\frac 1a\cdot \vec u$ (provided $a\ne 0$, of course).