If $\alpha(s)$ is a unit speed curve with $k\ne0$, how can we show that the equation of the osculating plane through $\alpha(0)$ is $[x-\alpha(0),\alpha'(0),\alpha''(0)] = 0$. (I mean 3 equal bars for the equal sign)
So what I'm thinking is we can use the fact that $[u,v,w] = [u\times v,w]$ and $k = T'/N$, Frenet serret doesn't look too helpful so I'm stuck. The definition of osculating plane is the plane $\alpha(s)$ perpendicular to $B$ (spanned by $T$ and $N$).