Suppose we have a smooth $k$-dim manifold $M \subset \mathbb{R}^$ and the tangent space at every point $p \in M$ (Here we translate every tangent space passing the origin point) has non-zero intersection with a given $(n-k)$-dim linear subspace $L$ in $\mathbb{R}^$.
Could we still find a smooth vector field locally (the vector field on $M$ is an assignment of a tangent vector to each point) on such that all vectors belong to $L$?
I know without the restriction that vectors belongs to $L$, the existence is not hard when thinking a vector field is a section of the tangent bundle. And if $$
I will appreciate for any useful answers and comments