Finally, one more rule. Not every set of 6 numbers corresponds to a line in 3D space. When solving problems using Pl cker coordinates, sometimes you end up with a line in Pl cker space, and you need to find the intersection of that line with the Grassman manifold to find the Pl cker points that correspond to the real solutions to the problem. More discussion of that particular operation is beyond the scope of this introduction, however. The striking similarity of the point and plane definitions tell us Pl cker 3D line coordinates precisely balance these extremes.
Many computational benefits accrue.
Also, many questions come in dual pairs, with the answers for one obtained by swapping U and V in the answers for the other. Even if you don't understand everything you read, it will give you a good overview of the other uses of Pl cker coordinates. That's probably enough for now. I hope this has been a gentle introduction to Pl cker coordinates. Let me know what you think about this introduction, and what you'd like to hear more about.