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The intersection of a line in Plücker coordinates with a plane covector $F$ is the first displayed homogeneous vector: it lies in the span of the defining points and its contraction with $F$ is zero by antisymmetry. A zero vector means the whole line is contained in the plane. Dually, the plane through the line and a point or ideal direction $D$ is given by the second formula; it vanishes if $D$ is already on the line. These operations require no Euclidean lengths or angles.

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.

Let $E_j$ denote the $j$th coordinate point. The plane through the given line and $E_j$ has covectorIndeed, 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 $X^0\ne 0$, 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.

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 areThis 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 ${\epsilon }_{0123}=1$ and define the dual Plücker coordinates byThe independent components pair asConversely, $L^{ij}=\frac{1}{2}{\epsilon }^{ijkl}{L}_{kl}$ 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, 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:This is a linear combination of $A,B$, so it is on the line, and $F_i{P}^i=0$ 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.