Here is a quotation from pg. 81, section 3-2 of Chern's Lectures on Differential Geometry:
Suppose $L^r=\{X_1,\ldots,X_r\}$ is a smooth $r$-dimensional distribution on $M$. [$M$ is an $m$-dimensional manifold.] ... Locally, the distribution $L^r$ is therefore equivalent to the system of equations $$ \omega_s = 0, \ \ \ \ r+1\le s \le m, $$ often called a Pfaffian system of equations.
(The $\omega_s$ are differential $1$-forms such that $\langle X_i, \omega_s\rangle = 0$ for all $i,s$.)
This just seems very odd to me. The $\omega_s$ are not $0$, as far as I can tell. They are a bunch of differential $1$-forms that annihilate the members of $L^r$, but they are not actually $0$. More precisely, for any $X\in T^*(p)$, $X\in L^r(p)$ if and only if $\langle X,\omega_s \rangle =0$ for all $s$. Right? So in what sense is it true that $\omega_s=0$?
In most of these lectures, Chern is extremely clear and careful. Is there a particular reason he is not here?
If I write the equation $x=0$, I do not mean that $x$ is identically zero; I am referring to its solution set. Similarly, Chern is saying that the distribution is defined by the equations $\omega_r=0$. In particular, the subspace consisting of all tangent vectors $X$ at a point annihilated by all the $1$-forms is precisely the distribution at that point. Moreover, in the case that the distribution is integrable, the pullbacks of the $\omega_r$ to an integral submanifold (i.e., their restrictions) will be $0$ (and conversely).