2. Both of your ideas are good, just go ahead: $\vec{u}\times\vec{v}$ is a vector orthogonal to both $\vec{u}$ and $\vec{v}$. Since the orthogonal complement of a plane in $\mathbb{R}^3$ is a line, $\vec{u}\times\vec{v}$ is a directing vector for the line you want. So a parametric equation for this line is: $\lambda\mapsto (2,3,0)+\lambda \vec{u}\times\vec{v}$, defined for $\lambda\in\mathbb{R}$.