I have solved a simple exercise, but in solving it I came to a contradiction. I want to know if it's correct.
In the exercise I have a curve parameterized by the arc length such that:
$\alpha (0)=\frac{1}{\sqrt{3}}(1, 1, 0)$
The binormal vector is $B(t)=\frac{1}{\sqrt{6}}(\sin t-\cos t,\sin t+\cos t,2)$
And the torsion is always negative. ($\tau(t)<0$)
I have to find the curvature of the curve.
I derived the binormal vector and then with the third equation of Frenet and knowing that the torsion is negative I determined the torsion (I took the norm from both sides). Knowing the torsion and using Frenet's third equation again, I could find the normal vector N. Then, I used $ N\wedge B = T $ to find the tangent vector. Finally I used the first Frenet equation to find the curvature. The problem is that when doing the whole process I got to the curvature is negative which is impossible. I want to know if something is wrong in my reasoning.