Skew Lines¶

Definition¶

Skew lines are two lines in $\mathbb{R}^3$ or higher dimensions that do not intersect and are not parallel.

Interpretation¶

Write the two lines in parametric form as $\boldsymbol{A}+t\boldsymbol{v}$ and $\boldsymbol{B}+s\boldsymbol{w}$$s\mathbf{w}$. They intersect only if these vectors are equal for some $s,t$; skewness requires that this system have no solution and that their directions not be parallel.

Skew lines show that higher-dimensional geometry has line relationships not possible in the plane.

The illustrated lines are skew: their directions are independent, but they have no common point. Skew lines can occur only in dimension at least $3$; in a plane, two non-parallel lines always meet.

Key Results¶

• Skew lines require dimension at least $3$.
• Two skew lines are neither parallel nor intersecting.

Examples¶

• The $x$-axis and the line through $(0,1,1)$ in direction $(0,1,0)$ are skew in $\mathbb{R}^3$.
• Two lines in $\mathbb{R}^2$ cannot be skew.

Prerequisites¶

• Parametric Representation of a Line

Related Concepts¶

• Plane
• Vector
• Normal Vector
• Euclidean Space

Sources¶

• Math II: Linear Algebra · Week 5

Aliases: non-coplanar lines, skew line Symbols: real numbers

Last updated: 20 October 2026 03:02