# Relationships between curvature, torsion, unit tangent vector, and binormal vector of a curve

Homework has already been collected and graded (but no explanation given) for these problems. I'm curious how to approach the problem.

Assume that the vector space we're in is $\Re^{3}$. Prove that

$$\begin{eqnarray*} (1) &\;\;\;\;\;\;\;\;& (\vec{\mathbf{\tau}} \cdot \vec{\mathbf{\beta}} \cdot \vec{\mathbf{\beta^{'}}}) &=& \kappa , \\ (2)&&(\vec{\mathbf{\beta}} \cdot \vec{\mathbf{\beta^{'}}} \cdot \vec{\mathbf{\beta^{''}}}) &=& \kappa^{2}(k / \kappa)^{'} ,\\ (3)&&(\vec{\mathbf{\tau}} \cdot \vec{\mathbf{\tau^{'}}} \cdot \vec{\mathbf{\tau^{''}}})&=& k^{3}(\kappa/k)^{'} , \end{eqnarray*}$$

where $\tau$ is the unit tangent vector, $\beta$ is the binormal vector, $\kappa$ is torsion, and $k$ is curvature. I started to attempt these proofs by starting from the vector form of the curve $$\vec{r}(t) = x(t)\vec{i} +y(t)\vec{j} +z(t)\vec{k}$$ and differentiating with respect to $t$ (and so on ...), but the algebra got really messy very quickly. Are there simpler relations between these mathematical objects that I'm missing or will I simply have to "grind out" the algebra?

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What is that dot product of three vectors you use in each of the equations? –  Henning Makholm Feb 10 '12 at 3:55
@Henning: Maybe the the box product (scalar triple product) was intended... Jubbles, have you looked up proofs for Frenet-Serret in textbooks? –  Guess who it is. Feb 10 '12 at 3:56
Maybe this? I've never seen the notation before, though. –  Dylan Moreland Feb 10 '12 at 3:57
Tau is the unit tangent while kappa is the torsion and k is the curvature? Holy switcharoo, Batman! That's confusing. –  anon Feb 10 '12 at 3:57
@anon: Correct. My professor has remarked that the textbook chose uncommon notation for curvature and torsion. –  Jubbles Feb 10 '12 at 4:00