I am trying to find a vector field on \begin{align} \mathbb{P}_\mathbb{R}^2 := \mathbb{R}^3 \setminus \{(0,0,0)\} \big/ \sim, \end{align} where $\sim$ is the equivalence relation for $z, z'\in \mathbb{R}^3$ defined by \begin{align} z \sim z' \iff z = \lambda z', \qquad \lambda \in \mathbb{R}\setminus {0}. \end{align}
Specifically, I am looking for a vector field that vanishes only at finitely many points.
I'm not sure how to start - I can't find any charts that make the answer apparent to me. I'd appreciate if anyone could explain how to go about this problem.