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?