# rotation vector

If $t(t),n(t),b(t)$ are rotating, right-handed frame of orthogonal unit vectors. Show that there exists a vector $w(t)$ (the rotation vector) such that $\dot{t} = w \times t$, $\dot{n} = w \times n$, and $\dot{b} = w \times b$

So I'm thinking this is related to Frenet-Serret Equations and that the matrix of coefficient for $\dot{t}, \dot{n}, \dot{b}$ with respect to $t,n,b$ is skew-symmetric.

Thanks!

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Suppose that $w=aT+bN+cB$, with $a$, $b$ and $c$ some functions. Then you want, for example, that $$\kappa N = T' = w\times T = (aT+bN+cB)\times T=b N\times T+cB\times T=-bB+cN.$$ Since $\{T,N,B\}$ is an basis, this gives you some information about the coefficients. Can you finish?
@Mariano: I got $\omega$ in terms of $\tau$ and $\kappa$, but the problem didnt specify a curve. How should I deal with this? –  ninja Sep 23 '11 at 20:00