Lines in space

Previously we came up with two algebraic representations for lines in 3D projective space: we got one from connecting two points and the other from intersecting two planes. In both cases the algebraic representation was an antisymmetric 4 × 4 matrix, but these two matrices were not the same. In this paper, we try to understand these differences and see how to convert one to the other. We become more acquainted with the properties of these matrices and build up some intuition on the geometric meaning of their components. We do all the algebra using what we call conventional, or familiar, notation. This gives letter names to each element of a vector/matrix and gives a sense of concreteness to the calculations. This makes the reader really appreciate the newer Einstein index notation and its alternate representation, tensor diagrams.