Chapter 11 — Three Dimensional Geometry

NCERT Class 12 Maths Chapter 11 covers Three Dimensional Geometry, teaching 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.

Chapter 11 of NCERT Class 12 Mathematics Part II introduces Three Dimensional Geometry using vector algebra to make the study simple and elegant. It covers direction cosines (cosines of angles a line makes with coordinate axes) and direction ratios (numbers proportional to direction cosines), with the key identity l² + m² + n² = 1. The chapter derives vector and Cartesian equations of lines in space, computes angles between two lines using the dot product formula, defines skew lines as non-parallel non-intersecting lines in different planes, and provides formulas for the shortest distance between skew lines and parallel lines using cross products.

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|