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Let's define $\mathbf a = \dot{\mathbf r},\ \mathbf b = \ddot{\mathbf r}$.$$
\frac{(\dot{\mathbf{r}}\times\ddot{\mathbf{r}}) \times\dot{\mathbf{r}}}{\lvert\dot{\mathbf{r}}
\rvert\lvert\dot{\mathbf{r}}\times\ddot{\mathbf{r}}\rvert}
$$
How do I take that dot product of the expression of above with itself?
I really need to get backLet's define $\mathbf a = \dot{\mathbf r},\ \mathbf b = \ddot{\mathbf r}$.
Then, if I understand you correctly, you're asking for:
$$((\mathbf a \times \mathbf b) \times \mathbf a)^2$$
According to the vector triple product we have:
$$(\mathbf a \times \mathbf b) \times \mathbf a = \mathbf b(\mathbf a \cdot \mathbf a) - \mathbf a (\mathbf a \cdot \mathbf b)$$
So:
$$((\mathbf a \times \mathbf b) \times \mathbf a)^2 = a^2b^2 - a^2 (\mathbf a \cdot \mathbf b)^2$$
It appears I simply do not understand your question.I really need to get back
$$
\frac{(\dot{\mathbf{r}}\times\ddot{\mathbf{r}})
\cdot\dddot{\mathbf{r}}}{ \lvert\dot{\mathbf{r}}\times
\ddot{\mathbf{r}}\rvert^2}
$$
afterwards though. I don't see how I can arrange your solution to do the job.
It appears I simply do not understand your question.
Perhaps you can clarify.
I see that document also provides the proof for the last theorem...It is related to proving the last theorem here:
http://www.math.wisc.edu/~seeger/234/frenet.pdf
which has to do with this questions here:
http://mathhelpboards.com/advanced-applied-mathematics-16/frenet-equation-torsion-6229.html
What is MIT?
MIT = Massachusetts Institute of Technology.What is MIT?
Assuming you mean the 2nd equality, they are using that:
\begin{aligned}
\mathbf T(s) &= \mathbf r'(s) \\
\mathbf T'(s) &= \kappa(s) \mathbf n(s) \\
\mathbf n(s) &= \frac{\mathbf r''(s)}{\kappa(s)} \\
\mathbf b(s) &= \mathbf T(s) \times \mathbf n(s) \\
\end{aligned}
Or if you meant the 3rd equality, it is explained on page 2 in your pdf.
Btw, can you please be more specific?
I dislike guessing what someone means.