I have the following situation: There is an embedded submanifold (in my particular case it is of codimension one but I do not think it matters here) $M\subseteq R^n$, and a vector subspace $P\subsetneq R^n$.
There are also a point $p\in P\cap M $ and a tangent vector point $v\in P\cap T_pM $. (Since $M$ is a submanifold of $R^n$ I am using the standard identification and considering $T_pM\subsetneq R^n$ so $P\cap T_pM $ makes sense).
It is also given that the intersection $P\cap M $ is a submanifold of $M$. My question is: does $v\in T_p(P\cap M) $?
My ideas so far are: $v \in P∩T_pM$ implies we can build two curves: $\alpha :I \mapsto P, \beta:I \mapsto M $ such that $\alpha (0)=\beta(0)=p, \alpha' (0)=\beta'(0)=v$. But I am not sure it is possible to find a single curve which passes throug $p$ with velocity $v$ that lies inside $P\cap M$.