Torsion formula derivation 
I am trying to derive the formula for torsion of a curve. In many example proofs I've seen the final step is that $\kappa=\|\dot{\lambda}\times\ddot{\lambda}\|$. However I thought curvature was defined to be $\kappa=\frac{\|\dot{\lambda}\times\ddot{\lambda}\|}{\|\dot{\lambda}\|^3}$ or $\kappa=\|\ddot{\lambda}\|$ if $\lambda$ is unit-speed. However, I would like to note, I have only assumed that $\lambda$ is a regular curve with non-zero curvature.
Edit: essentially how does one derive the formula in the screenshot.
 A: We must use the Frenet formulas for regular curves:
$$
\begin{align*}
T' &= v\kappa N\\
N' &= -v\kappa T + v \tau B\\
B' &= -v\tau N.
\end{align*}
$$
Here $T, N, B$ is the Frenet frame along the curve $\gamma$, $v = \|\gamma'\|$ is the speed and $\kappa$ and $\tau$ are of course the curvature and torsion.
By definition of $v$ we have
$$
 \gamma' = vT.
$$
The second derivative of $\gamma$ is
$$
 \gamma'' = v'T + vT' = v' T + v^2\kappa N.
$$
The cross product $\gamma'\times \gamma''$ becomes
$$
 \gamma' \times \gamma'' = \kappa v^3 B.
$$
If we take the norm of both sides, we obtain
$$
 \kappa = \frac{\|\gamma'\times\gamma''\|}{v^3} = \frac{\|\gamma'\times \gamma''\|}{\|\gamma'\|^3}. \tag{1}
$$
The binormal vector $B$ is a unit vector perpendicular to $T$ and $N$. From the expressions for $\gamma'$ and $\gamma''$, it follows that $B$ points in the same direction as $\gamma'\times \gamma''$, so
$$
 B = \frac{\gamma'\times\gamma''}{\|\gamma'\times\gamma''\|}. \tag{2}
$$ 
Deriving one more time gives $\gamma'''$:
$$
\begin{align*}
 \gamma''' &= v'' T + v'T' + (v^2\kappa)'N + v^2\kappa N' \\
          &= v''' T + (v'v\kappa  + (v^2\kappa)')N + v^3\kappa (-\kappa T + \tau B).
\end{align*}
$$
Take the inner product with $B$ and obtain $\gamma'''\cdot B = v^3\kappa \tau$. Combined with $(2)$ this gives
$$
 \tau = \frac{\gamma''' \cdot \gamma'\times\gamma''}{\|\gamma'\times\gamma''\|v^3\kappa}.
$$
Finally, using $(1)$, we may conclude that the torsion is given by
$$
  \tau = \frac{\gamma'\times\gamma''\cdot\gamma'''}{\|\gamma'\times\gamma''\|^2}.
$$
