Chapter 11 — Three Dimensional Geometry

Chapter 11 of the Class 12 Maths NCERT textbook, "Three Dimensional Geometry", teaches direction cosines and direction ratios of lines, vector and Cartesian equations of lines and planes, angle between two lines, and shortest distance between skew and parallel lines using vector algebra.

• Fixing direction in space — The chapter captures the orientation of a line through direction cosines — the cosines of the angles it makes with the axes — and the proportional direction ratios, anchored by the identity l² + m² + n² = 1 that keeps direction consistent in three dimensions.
• Equations of lines in 3D — Using vector algebra, lines are written in both vector form (r = a + λb) and equivalent Cartesian form, making 3D geometry compact and elegant. From these, the angle between two lines follows directly from the dot product of their directions.
• Distance between skew lines — The chapter defines skew lines — neither parallel nor intersecting — and tackles the harder question of how far apart they are, measuring the shortest distance along the common perpendicular using the cross product of their direction vectors.

1. 01Direction cosines l, m, n of a line satisfy l² + m² + n² = 1; direction ratios a, b, c are any numbers proportional to l, m, n
2. 02Direction cosines of the line joining P(x₁,y₁,z₁) and Q(x₂,y₂,z₂) are (x₂−x₁)/PQ, (y₂−y₁)/PQ, (z₂−z₁)/PQ where PQ is the distance between P and Q
3. 03The vector equation of a line through point with position vector a and parallel to vector b is r = a + λb; the Cartesian form is (x−x₁)/a = (y−y₁)/b = (z−z₁)/c
4. 04Two lines with direction ratios a₁,b₁,c₁ and a₂,b₂,c₂ are perpendicular when a₁a₂+b₁b₂+c₁c₂ = 0, and parallel when a₁/a₂ = b₁/b₂ = c₁/c₂
5. 05Skew lines are lines in space that are neither parallel nor intersecting; the shortest distance between them is along the line perpendicular to both
6. 06Shortest distance between two skew lines r = a₁+λb₁ and r = a₂+μb₂ is |(b₁×b₂)·(a₂−a₁)| / |b₁×b₂|