Skew lines and what's between them Is it always possible to find a line perpendicular to two skew lines in space?
And how can we visualise the proof geometrically? And if anyone could present the proof that it is always possible to exist a line perpendicular to both skew lines, please elaborate.

If you like to use visualization and manipulatives, hold a pair of pencils as your two skew lines.  Pick any random orientation for each pencil.  Visualize a line segment that represents the closest approach between the pencils.
Is the closest approach path perpendicular to both pencils?  Keeping the two pencils in a fixed juxtaposition to each other, rotate the ensemble so that both pencils are level to the ground.  This special case of skew lines demonstrates that the skew lines lie in two parallel planes, and the closest approach runs perpendicular to these planes.
Since angles are preserved under 3D rotation, this same relationship where the closest approach runs perpendicular to both skew lines must apply to all pairs of skew lines, since they are just 3D rotations of the special case.
