How do we explain to students that division by a vector does not make sense?

Question

I have to give a presentation on vector analysis. One of many important things I want to emphasize is that a division by a vector does not make sense.

How do you explain to your students, for example, that division by a vector does not make sense?

Bonus question: Also how do you explain that integration with $dx$ in the denominator does not make sense? Consider the following $\int f(x)\frac{1}{\textrm{d}x}$.

Answer 1

Division should be the inverse of multiplication. What is multiplication for vectors? You could multiply a vector by a scalar obtaining a vector, or we could multiply two vectors to obtain a scalar.

In the first case on could possibly define division... but it would be defined only in the very restrictive case of a vector being a multiple of the other. In the second case (scalar multiplication) division would not be well defined because there are many vectors which multiplied to a given vector give the same result.

As for the second question... you cannot explain all things that do not make sense. You define some things, and that's all. Otherwise you should explain why we don't define untegrals or why we don't define $\sqrt{\int^\partial}$.

Don't let your students think that everything that can be written could possibly make sense...

Answer 2

It depends in which way you want to define the product of two vectors. Clearly if you think a vector in $\mathbb R^2$ as a complex number, then $\mathbb R^2$ is a field.