Skew Lines - Explanation

Explanation

If each line in a pair of skew lines is defined by two points, then these four points must not be coplanar, so they must be the vertices of a tetrahedron of nonzero volume; conversely, any two pairs of points defining a tetrahedron of nonzero volume also define a pair of skew lines. Therefore, a test of whether two pairs of points (a,b) and (c,d) define skew lines is to apply the formula for the volume of a tetrahedron, V = |det(a−b, b−c, c−d)|/6, and testing whether the result is nonzero.

If four points are chosen at random within a unit cube, they will almost surely define a pair of skew lines, because (after the first three points have been chosen) the fourth point will define a non-skew line if, and only if, it is coplanar with the first three points, and the plane through the first three points forms a subset of measure zero of the cube. Similarly, in 3D space a very small perturbation of two parallel or intersecting lines will almost certainly turn them into skew lines. In this sense, skew lines are the "usual" case, and parallel or intersecting lines are special cases.