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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?2012-02-10
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    @Henning: Maybe the the box product (scalar triple product) was intended... Jubbles, have you looked up proofs for Frenet-Serret in textbooks?2012-02-10
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    Maybe [this](http://en.wikipedia.org/wiki/Triple_product)? I've never seen the notation before, though.2012-02-10
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    Tau is the unit tangent while kappa is the torsion and k is the curvature? Holy switcharoo, Batman! That's confusing.2012-02-10
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    @anon: Correct. My professor has remarked that the textbook chose uncommon notation for curvature and torsion.2012-02-10
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    If you've learned about [Frenet](http://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas) formulae (like J.M. asked), all of them are (almost) direct consequences.2012-02-10
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    @J.M.: You are correct (also credit to Dylan). It is the scalar triple product (mixed or box product). I am discovering that the text for my class (ISBN 0817643842) uses different notation than most other differential geometry books. Also, it has tons of typos.2012-02-13

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