# Orthogonal derivative of binormal vector.

1. Oct 19, 2011

### Locoism

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

Suppose that r(s) defines a curve parametrically with respect to arc length and the r′(s) is nonzero on the curve. Show that dB/ds is orthogonal to both
B(s) and T(s). Conclude that there is a scalar function τ(s) such that
dB/ds = −τ (s)N . (This function τ is known as the torsion of the curve.)

3. The attempt at a solution
Not really sure how to approach this one. I'd say we need to show some way that dB/ds x B = 0 and the same with T, but I'm not sure how to do it completely algebraically... As for τ, is it enough just to show that dB/ds is parallel to N? This would simply follow from the proof that it is orthogonal to B and T, since T x B = N (or T x N = B)