green knot
Projective geometric algebra is the plane based algebra, where 1-vectors correspond to planes. e_1 is the plane which has the ordinary (1,0,0) as a normal and e_0 is the plane at infinity. Calculating by hand and wanting to solve for the line which goes through the point (2,0,3) and has a direction e_13, I used the Meet operator, which is just the wedge product.
2e_1 ^ 3e_3 = 6e_13, the bivector that goes through... the origin? Then I realized that I had made a beginner mistake and instead I would need to use the plane at infinity which is acts as a translation and takes a plane "from the origin" to the direction of the normal. So e_1+2e_0 is the plane where x=2 and y, z are free.
So now the exterior product for a line that would go through (2,0,3) is:
(2e_0+e_1) ^ (3e_0 + e_3) = 6 e_0e_0 + 2e_03 + 3e_10 + e_13. e_0e_0 = 0 and we are left with the bivector component which is correct and then the translation components. The translation seems correct, but 3e_10 is wrong, or at least feels wrong as the "standard" base is the base with indices in ascending order, so e_01 = -e_10.
Not wanting to struggle too much with this, the line is then the bivector (0, 1, 0) + translative bivector (2,0,3). The basis e_13 = -e_31 can be read "not 1st, not 3rd", but what about the translations? What about lines at infinity (translations), the exterior product of v = 4e_1 + 2e_0 and u = e_1 + e_0 is 4e_10 + 2_e01 = 2e_10, even though the translational difference is only 1.
torn moon