Skew LinesJEE Advanced

Concept

Skew lines are lines in 3D that neither meet nor run parallel. Their shortest distance is measured along the unique common perpendicular, and the cross product of their direction vectors points exactly along that perpendicular.

Key formula

Derivation

Let the lines be $\vec r = \vec a_1 + t\vec b_1$ and $\vec r = \vec a_2 + s\vec b_2$. The common perpendicular direction is $\vec b_1\times\vec b_2$.

Project the connecting vector $\vec a_2 - \vec a_1$ onto that unit perpendicular: the magnitude of this projection is the shortest distance.

Hence $d = |(\vec a_2 - \vec a_1)\cdot(\vec b_1\times\vec b_2)| / |\vec b_1\times\vec b_2|$. Sliding either line along itself changes $\vec a_i$ but not the result.

Scenarios to explore

• Skew Lines — Shortest distance between two skew lines in 3D.

Real-world applications

• Clearance between non-intersecting pipes, cables or robot arms.
• Closest approach of two moving particles with constant velocities.
• Computer-graphics collision and proximity tests.

JEE exam tips

• If $(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2) = 0$, the lines are coplanar (they intersect).
• Parallel lines need a different formula: $d = |(\vec a_2-\vec a_1)\times\vec b|/|\vec b|$.

Common mistakes

• Forgetting the absolute value in the numerator (distance is non-negative).
• Using $|\vec b_1||\vec b_2|$ instead of $|\vec b_1\times\vec b_2|$ in the denominator.
• Assuming the distance changes when you slide a line along its own direction.

Exam traps to avoid

• Zero scalar triple product means coplanar, not necessarily intersecting at the origin.
• The numerator is the scalar triple product $[\vec a_2-\vec a_1\;\vec b_1\;\vec b_2]$.