And in 3 dimensions it's points and planes that are dual - and finding the line between two points or the common line of two planes can now be done in a similar way, but using Plücker coordinates for the lines instead.
Those two operations are called "meet" and "join", and in principle they work with any linear subspaces (point, line, plane, etc.) So the meet of a line and a plane in 3D is a point, the join of a point and a line in 3D is a plane, etc.
Wikipedia is good on this stuff if you are patient with mathematician's way of expressing things...
The operations intersection and sum make the set of all subspaces a bounded modular lattice, where the {0} subspace, the least element, is an identity element of the sum operation, and the identical subspace V, the greatest element, is an identity element of the intersection operation."
Wikipedia is good on this stuff if you are patient with mathematician's way of expressing things...
https://en.m.wikipedia.org/wiki/Join_and_meet
https://en.m.wikipedia.org/wiki/Linear_subspace
"Lattice of subspaces
The operations intersection and sum make the set of all subspaces a bounded modular lattice, where the {0} subspace, the least element, is an identity element of the sum operation, and the identical subspace V, the greatest element, is an identity element of the intersection operation."