Plane intersecting all the lines

Question

This might sound a bit stupid or ill thought, but I am having trouble visualizing it and proving it.

Given a finite set $L$ of straight lines in $\mathbb{R^3}$ is it always possible to find a plane which intersects all the lines ?

Answer 1

A given plane intersects a given line unless the normal-vector of the plane is perpendicular to the line.

Given a finite set of straight lines, for each line find the plane that is perpendicular to that line and goes through the origin. At the end you have a finite number of planes all passing through the origin. Pick any vector that does not go through those planes and pick a plane perpendicular to that vector. That plane will intersect all the lines.

Answer 2

A line and a plane intersect iff the plane's normal vector isn't orthogonal to the line's direction. Given a finite set of lines, just choose as normal vector any vector that isn't orthogonal to any direction of the lines.