# Ruled surface swept out by normal lines of a curve

The problem is: "Say $\gamma:[a,b] \to \mathbb{R}^3$ is a curve of general type with principal normal vector field $\textbf{n} = t_2$. Show that the ruled surface $r(u,v) = \gamma(u) + v\textbf{n}(u)$ is developable if and only if $\gamma$ is planar."

I was going to compute the first and second fundamental forms, find the Weingarten matrix, then take the determinant and hope it's 0 iff $k_2 = 0$ or something, but this seems tedious. Is there a better way I shoud be doing this?