Plücker coordinates ($L^{ij}=A^i{B}^j-B^i{A}^j$)

A line in real projective space ${\mathbb{P}}^3$ is the span of two independent homogeneous vectors $A,B$. Its Plücker coordinates are the entries of the displayed rank-two antisymmetric matrix, determined up to a nonzero scalar. They satisfy $L^{01}{L}^{23}-L^{02}{L}^{13}+L^{03}{L}^{12}=0$. With a chosen volume orientation, its dual plane form is $L_{ij}=\frac{1}{2}{\epsilon }_{ijkl}{L}^{kl}$. This is projective duality using the volume form, not lowering indices with a Euclidean metric.

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.