# Homework Help: Unit Normal Vector N(t)

1. Dec 7, 2007

### issisoccer10

1. The problem statement, all variables and given/known data
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)||

2. Relevant equations
T(s) = r'(s)
||r'(s)|| = 1

3. 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

2. Dec 7, 2007

### Dick

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

3. Dec 7, 2007

### 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.