I'm trying to solve the following problem:
Let $\tilde M$ be the set of real matrices $3 \times 2$ of rank 2.
$$ (u,v) = \begin{pmatrix} u_1 & v_1 \\ u_2 & v_2 \\ u_3 & v_3 \end{pmatrix}$$
and $M$ the set of the vectorial planes of $\mathbb{R}^3$
Give a differentiable structure to $M$ to make the map $\pi: \tilde M \to M$ which sends $(u,v)$ to $L(u,v)$ a submersion.
I have no idea of what chart I have to choose. I have already proven that $\tilde M \subset \mathbb{R}^6$ is an open set so I just have to find a suitable atlas for $M$, calculate the differential $\pi_*$ (computing the derivatives with respect to the $u_i$ and $v_i$) and check that it is surjective.
What charts should I choose? I don't know how to find a suitable atlas for $M$.
Any plane in $\mathbb R^3$ determines an orthogonal line. This gives a bijection between $M$ and $\mathbb {RP}^2$, the projective plane. The atlas on $M$ is determined by this map and the atlas on the projective plane.
A good resource for the smooth structure on $\mathbb {RP}^2$ is given in Lee's Smooth Manifolds.