Given the following three lines along the edge of a unit cube, describe all lines that meet all three of these.

Question

My attempt: Parametric equations for the three lines are given by

(x,y,z) = (t,0,0)

(x,y,z) = (0,s,1)

(x,y,z) = (1,1,r)

Any line meeting all three of these must pass through a point satisfying each of the above equations, but I don't know how to go about calculating this.

More generally, some intuition behind using parameterisations of planes and lines to find their intersection would be appreciated!

Answer 1

The lines are parallel to (x,y,z) axes, but are pairwise skew. Skew lines by definition do not meet, and the three lines have no intersecting lines for any of the three pairs.