I was rifling through some USSR problem books (with no solutions!) and I found this one, its really easy, but I was dissatisfied with my proof. I think it can be done with less work.

Prove that the principal normal unit vector, denoted does not exist for a linear function

My proof is, dont look if you dont want to

so

and

So the tangent unit vector is

So now

and

does not exist because there will be a division by zero