Skew Lines

Definition. Two straight lines in space are called skew lines if they satisfy both of the following conditions simultaneously: (i) they are not parallel (their direction vectors are not scalar multiples of each other), and (ii) they do not intersect (there is no point common to both lines). A pair of lines that fails either condition is not skew: two lines with proportional direction vectors are parallel (whether or not they coincide), and two lines with non-proportional direction vectors that do share a common point are ordinary intersecting lines.

Why skew lines cannot exist in a plane. In plane (2D) geometry, any two distinct straight lines are related in only one of two ways: either their direction vectors are proportional, making them parallel, or their direction vectors are not proportional -- and in a plane, two lines with genuinely different directions are always forced to cross somewhere, so they must intersect. Skew lines are therefore a genuinely three-dimensional phenomenon: it is only because space has a third direction to move in that two non-parallel lines can be positioned so that they pass by each other without ever meeting -- for instance, one line lying entirely in the plane $z=0$ and a second, non-parallel line lying entirely in a different plane $z=k$ ($k\ne0$), can never share a point, since every point of the first has $z=0$ and every point of the second has $z=k\ne0$.

Testing whether two lines are skew. Given two lines in vector form $\vec r=\vec a_1+\lambda\vec b_1$ and $\vec r=\vec a_2+\mu\vec b_2$, determining their relationship is a two-step check:

1. Check for parallelism first. Compare the direction vectors $\vec b_1$ and $\vec b_2$: if one is a scalar multiple of the other, the lines are parallel (and skew is immediately ruled out).
2. If not parallel, check for intersection. Equate the general points, $\vec a_1+\lambda\vec b_1=\vec a_2+\mu\vec b_2$, component-wise, giving three scalar equations in the two unknowns $\lambda,\mu$. Solve any two of the three equations simultaneously, then substitute both values into the third: if it is satisfied, a genuine common point exists, so the lines intersect; if it is not satisfied, no common point exists for any choice of parameters, so the lines are skew.

This test is purely algebraic: three equations in two unknowns are, in general, an overdetermined system, and whether or not it happens to be consistent is exactly what decides between intersecting and skew. …