Consider two vector fields $F_1,F_2$ of $\mathbb{R}^3$ and define the distribution $$D(q) =\operatorname{span}\left\{F_1(q),F_2(q)\right\}=\left\{a(q)F_1(q)+b(q)F_2(q),\; a,b \in \mathcal{C}^\infty\right\}$$
Suppose $q=(x,y,z)$ such that $$ \det(F_1(q),F_2(q),[F_1,F_2](q))=0. $$
Why is there $\alpha$ such that $D$ is isomorphic to $\ker \; \alpha$ and $\alpha = y\, dx +dz$ ?