I am trying to prove the following expression:
By expanding the RHS of the expression, show that $$ \frac{d\hat{T}}{dt}=\frac{r'\times(r''\times r')}{{\Vert r\Vert}^3} $$
where $\hat T$ is the unit tangent vector to a curve, i.e. $\hat T=\frac{r'}{\Vert r'\Vert}$
I used the vector triple product to find the numerator which gives me $r''(r'\cdot r')-r'(r'\cdot r'')$ I know that $r'\cdot r'={\Vert r'\Vert}^2$, but I do not know how to simplify the second bracket.
Does anyone have any suggestions or could push me in the right direction?
$$ {d\hat T\over dt}= {d\over dt}{r'\over\sqrt{r'\cdot r'}}={r''\over\sqrt{r'\cdot r'}} -{r'\over2}{2r'\cdot r''\over(r'\cdot r')^{3/2}}= {r''(r'\cdot r')-r'(r'\cdot r'')\over(r'\cdot r')^{3/2}}. $$