Geometry Problem: Does a Plane Exist Where p Non-Intersecting Lines Meet?

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In R^3, the discussion centers on whether a plane exists where p non-intersecting lines can all intersect that plane. The original question raises curiosity about the geometric properties of lines in three-dimensional space. A follow-up question considers the homeomorphism between R^3 minus n non-intersecting lines and R^3 minus n parallel non-intersecting lines. The conversation invites insights into the implications of these geometric configurations. Overall, the thread explores intriguing concepts in geometry related to lines and planes in three-dimensional space.
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Homework Statement


In R^3, if we have p non-intersecting lines, is it true that there exists a plane Y such that all the lines intersect Y? Up to rotations.

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The Attempt at a Solution



No real attempt, this is just something I found to be interesting. Any thoughts?
 
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Here is a better question:

Is R^3 - {n non-intersecting lines} homeomorphic to R^3 - {n parallel non intersecting lines}?
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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