= 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 to
$$
\boxed{P_{(j)}^i=L^{ij}\qquad\text{(no sum over }j\text{)}.}
$$
Thus 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.
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