= Solution
Let $E_j$ denote the $j$th coordinate point. The plane through the given line and $E_j$ has covector
$$
\boxed{F_i^{(j)}=L_{ij}\qquad\text{(no sum over }j\text{)}.}
$$
Indeed, the lower-index <Plücker coordinates> annihilate both line-defining points, while $F_i^{(j)}E_j^i=L_{jj}=0$. The $j$th column therefore lists this plane's coefficients. In the affine chart $X^0\ne0$, the points $E_1,E_2,E_3$ lie at infinity and represent the three coordinate axis directions; a plane containing $E_j$ is parallel to that axis direction. If $E_j$ already lies on the line, the column is zero and infinitely many planes through the line contain that direction. This is the dual of the containment degeneracy in the coordinate-plane intersection.
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