 Solution

The $j$th coordinate plane has equation $X^j=0$, hence covector $F_i={\delta }_{ij}$. The line-plane incidence in Plücker coordinates formula reduces toThus the $j$th column of the upper-index Plücker coordinates contains the intersection point's homogeneous coordinates, up to projective scaling. Its $j$th entry vanishes as required. If that column is zero, both defining points have coordinate $j$ zero and the entire line is contained in the coordinate plane; there is no distinguished intersection point. Reading rows instead changes only the irrelevant overall sign.