= Solution
Work in three-dimensional <projective space>, with four <homogeneous coordinates> indexed $0,1,2,3$. Let $A$ and $B$ be independent representatives of two points. The <Plücker coordinates> of their line are
$$
\boxed{L^{ij}=A^iB^j-B^iA^j.}
$$
This <antisymmetric matrix> has rank two and is defined only up to a nonzero scalar. Replacing the pair by another <basis> of the same two-dimensional subspace multiplies $L$ by the determinant of that basis change. Its six independent entries satisfy $L^{01}L^{23}-L^{02}L^{13}+L^{03}L^{12}=0$, since the <exterior product> $(A\wedge B)\wedge(A\wedge B)$ vanishes.
To describe the same line as an intersection of planes, choose a volume orientation with $\varepsilon_{0123}=1$ and define the dual <Plücker coordinates> by
$$
\boxed{L_{ij}=\frac12\varepsilon_{ijkl}L^{kl}.}
$$
The independent components pair as
$$
L_{01}=L^{23},\quad L_{02}=-L^{13},\quad L_{03}=L^{12},\quad
L_{12}=L^{03},\quad L_{13}=-L^{02},\quad L_{23}=L^{01}.
$$
Conversely, $L^{ij}=\tfrac12\varepsilon^{ijkl}L_{kl}$ with the matching volume convention. The lower-index form annihilates both defining points: $L_{ij}A^j=L_{ij}B^j=0$. Its columns span the pencil of planes containing their line. This dualization uses the volume form, \b[not a Euclidean metric for raising and lowering indices]. Two-index point and plane representations are dual in projective dimension three; in higher dimensions the corresponding dual exterior form has a different degree.
For a general plane covector $F_i$, contract the upper-index form:
$$
\boxed{P^i=L^{ij}F_j=A^i(F_jB^j)-B^i(F_jA^j).}
$$
This is a <linear combination> of $A,B$, so it is on the line, and $F_iP^i=0$ by antisymmetry, so it is on the plane. Provided the resulting vector is nonzero, its <homogeneous coordinates> give the unique intersection, including intersections at infinity. If $F(A)=F(B)=0$, the whole line lies in the plane and the zero result correctly indicates that there is no unique intersection. These are the <line-plane incidence in Plücker coordinates> operations used in the subparts.
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