I'm tryig to prove the following question. Let $M$ be a surface and $\alpha$ be a unit speed curve in M.
(1) $\alpha$ is both principal and geodesic iff it lies in a plane everywhere orthogonal to $M$ along $\alpha$.
(2) $\alpha$ is both principal asymptotic iff it lies in a plane everywhere tangent to $M$ along $\alpha$.
My thoughts: I know that if U is a unit normal vector field on M then $\alpha$ is principal implies that $\alpha'$ is collinear to U'. $\alpha$ is geodesic implies $\alpha''$ is normal to $M$. $\alpha$ is asymptotic implies $\alpha''$ is tangent to M. How do I use this information to say that $\alpha$ lies in a certain plane?
Any help is appreciated.
HINTS:
(1) If $\alpha$ is geodesic, as you've said, $\alpha''$ is parallel to $U$. So what is the derivative of $U\times\alpha'$?
(2) If $\alpha$ is principal, then $U' = k\alpha'$ for some function $k$ along $\alpha$; if $\alpha$ is an asymptotic curve, what is $k$?