Skew gap
"Skew gap" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Skew lines and the shortest distance" in Three Dimensional Geometry. Blue beam 1 runs from F(1, 2, 0) along 2î + 3ĵ + 6k̂; green beam 2 passes A(3, 6, 2) along 6î + 2ĵ − 3k̂. They are not parallel and never meet: they are skew. P slides t metres along beam 1 from F, Q slides s metres along beam 2 from A. The orange segment PQ and its length change as you slide. The shortest segment between skew lines is perpendicular to both, so it points along b₁ × b₂. Its length is the shadow of any joining vector a₂ − a₁ on that direction: d = |(b₁ × b₂)·(a₂ − a₁)|/|b₁ × b₂|. JEE Main asked 84 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Skew lines and the shortest distance in Three Dimensional Geometry. Change the values and watch what happens.
In JEE Main
42 shifts analysed
84 questions from Three Dimensional Geometry across the 42 JEE Main shifts Lumi has analysed (2017–2026).
The idea behind it
NCERT §11.5–11.5.1
- Two lines in space either meet, are parallel, or are skew: neither parallel nor meeting, so no plane holds both. In a box-shaped room, a diagonal of the ceiling and a diagonal of a side wall can be skew.
- The shortest distance between two lines is the length of the shortest segment joining a point of one to a point of the other. It is 0 for meeting lines.
- For skew lines the shortest segment is perpendicular to both lines, so it runs along b₁ × b₂.
- For r = a₁ + λb₁ and r = a₂ + μb₂ the shortest distance is d = |(b₁ × b₂)·(a₂ − a₁)| / |b₁ × b₂|: the projection of any joining vector a₂ − a₁ on the common perpendicular.
- With coordinates, the top is a determinant: stack the joining vector a₂ − a₁ above the two ratio triples and take its absolute value. The bottom is |b₁ × b₂|, the length of the cross product of the ratio triples.
- Worked: r = î + ĵ + λ(2î − ĵ + k̂) and r = 2î + ĵ − k̂ + μ(3î − 5ĵ + 2k̂). Here a₂ − a₁ = î − k̂ and b₁ × b₂ = 3î − ĵ − 7k̂ of length √59, so d = |3 + 7|/√59 = 10/√59.
- If the numerator is 0 (and the lines are not parallel), the lines meet: that determinant being 0 is the test for two lines to intersect.
Questions
3 answers
- What does the "Skew gap" simulation show?
- Blue beam 1 runs from F(1, 2, 0) along 2î + 3ĵ + 6k̂; green beam 2 passes A(3, 6, 2) along 6î + 2ĵ − 3k̂. They are not parallel and never meet: they are skew. P slides t metres along beam 1 from F, Q slides s metres along beam 2 from A. The orange segment PQ and its length change as you slide.
- Does JEE Main ask questions from Three Dimensional Geometry?
- Yes. JEE Main asked 84 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Skew lines and the shortest distance" in Three Dimensional Geometry (Class 12 Maths).
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