I have been presented with the problem where I have to prove that for a curve $C$ that is parameterized by $\gamma (s)$, where are $\gamma (s)$ has unit-speed, is spherical, i.e. every point of $C$ lies on the sphere $\mathbb{R}^3$. We also suppose that $\gamma (s)$ is spanned by a normal vector $n(s)$ and the binormal vector $b(s)$ for every $s$ that passes through a given fixed point $x_0$ in $\mathbb{R}^3$.
I have only begun my study of differential geometry so I am quite stuck as to where to start the proof. But, I do know that for $\gamma (s)$ to be unit-speed: $|| \dot \gamma(s) || = 1$. Also, that $n(s) = \frac{\ddot \gamma (s)}{\kappa (s)}$ and $b(s) = t(s) \times n(s)$, where $t(s) = \dot \gamma (s)$.
Any and all help would be greatly appreciated!
Hint: The condition $\gamma$ is spanned by $n$ and $b$ means that
$$t(s)\cdot\gamma(s) = 0$$
for all $s$.