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(By a complex position I mean a position vector whose Cartesian coordinates are complex.)

I guess the simple answer is “no”, as any intersections (of which there can be none, one, or infinitely many) are real.

But I am wondering if there is some set of equations which is solved by a real position for intersecting lines, and by a complex position for skew lines(?).

Ideally, that complex position has some meaning. For example, if P and Q are the points on the two lines that are closest to the other line, it would be magic if the real part of that complex position is (P+Q)/2.

jcuk
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  • Interesting question. I have no answer. In case it has some interest for you, here is a good (and deep) paper on skew lines. – Jean Marie Mar 21 '21 at 14:10
  • Thank you, @MorganRodgers. I will try what you suggest. I think I somehow need to create a quadratic equation to get complex solutions... maybe your approach works. Thanks again! – jcuk Mar 22 '21 at 09:25
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    Thanks very much, @JeanMarie. I will complete the "Light reading for the professional" task you set me :) Thanks again! – jcuk Mar 22 '21 at 09:27
  • I will think again to it. It should be possible in the spirit of a question I have asked 2 years ago. – Jean Marie Mar 22 '21 at 09:46

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