Why Is the Unit Normal Vector N(s) Defined as r''(s)/||r''(s)||?

[issisoccer10]

Homework Statement

This question comes from my multivariable differential calculus course, and it pertains to arc length parametrizations the unit normal vector of a curve.

Why does the arc length parametrization of the unit normal vector of a curve, N(s) equal...

_r''(s)__ = N(s) and not just equivalent to r''(s) = N(s)?
||r''(s)||

Homework Equations

T(s) = r'(s)
||r'(s)|| = 1

The Attempt at a Solution

I thought that since ||r'(s)|| = 1, ||r''(s)|| would be equivalent to 1 as well since they are both the normalizations of arc length parametrizations of curves. However, this apparently isn't the case...any help would be appreciated

thanks

 

[Dick]

Well, take a 'fer instance'. r(s)=(1/2)*(cos(2s),sin(2s)). r'(s) is unit length, r''(s) isn't.

 

[HallsofIvy]

I'm just seconding Dick. If a curve is paraemtrised by arclength \vec{r}(s), then it follows from the definition of "arclength", and the chain rule, that the length of \vec{r}' is 1. There is no reason to expect that to be true for \vec{r}'' as well.

Of course, \vec{r}'' is normal to the curve. I just doesn't have length 1.