Plücker coordinates

Remarks

The $p_{ij}$ are the $2 \times 2$ minors of the matrix with rows $x$ and $y$. If $x, y$ are replaced by another basis $x' = a x + b y$, $y' = c x + d y$ of the same plane, each $p_{ij}$ is multiplied by the determinant $ad - bc \ne 0$, so the point of $\mathbb{P}^5(F)$ depends only on $L$. The coordinates satisfy the Plücker relation $$p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12} = 0.$$ In the language of the exterior algebra, the $p_{ij}$ are the coordinates of $x \wedge y$ in the basis $e_i \wedge e_j$ of $\Lambda^2 F^4$.

Example: for $x = (1, 0, 0, 0)$ and $y = (0, 1, 0, 0)$, one gets $p_{01} = 1$ and all other $p_{ij} = 0$, so the line has Plücker coordinates $[1 : 0 : 0 : 0 : 0 : 0]$, and the relation holds since every product in it is $0$.

Non-example: if $y = 2x$, every $p_{ij}$ is $0$, which is not a point of $\mathbb{P}^5(F)$; the vectors must be linearly independent to span a line.