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Three Dimensional Geometry

NCERT Class 12 Mathematics Chapter 11

Three Dimensional Geometry for Class 12 Maths and JEE: direction cosines and ratios, the equation of a line, angle between lines and shortest distance.

5 topics Updated 8 Nov 2026

For: Class 12 and JEE, unless a topic says otherwise

In the syllabus: JEE Class 12 syllabus

With your textbook

NCERT Mathematics Part II (Class 12), Chapter 11 “Three Dimensional Geometry”, reprint 2026-27.

Open the official NCERT chapter (PDF, 2.1 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Direction cosines and direction ratios of a line

    Direction cosines and direction ratios for Class 12 Maths and JEE: l + m + n = 1, cosines from ratios, two points, collinear points, solved examples.

    NCERT Section 11.2 10 min read 5 solved examples 10 practice questions

  2. Equation of a line in space: vector and Cartesian forms

    Equation of a line in 3D for Class 12 Maths and JEE: vector form r = a + λb, parametric and Cartesian forms, the line through two points, solved examples.

    NCERT Section 11.3 11 min read 5 solved examples 11 practice questions

  3. Angle between two lines in space

    Angle between two lines in 3D for Class 12 Maths and JEE: cos θ from direction ratios, perpendicular and parallel lines, vector form, solved examples.

    NCERT Section 11.4 11 min read 6 solved examples 11 practice questions

  4. Shortest distance between two lines: skew and parallel lines

    Shortest distance between two lines for Class 12 Maths and JEE: skew lines, the cross-product formula, Cartesian form, parallel lines, NCERT questions.

    NCERT Section 11.5 12 min read 5 solved examples 11 practice questions

  5. Line of shortest distance between two skew lines

    The common perpendicular, its feet from two dot products, and its equation, with checks.

    JEE only 9 min read 3 solved examples 8 practice questions

Key formulas in this chapter

Direction cosines and ratios

  • l + m + n = 1
  • l = ±a/√(a + b + c), and the same for m, n
  • through P, Q: cosines (x − x)/PQ, (y − y)/PQ, (z − z)/PQ

Equation of a line in space

  • r = a + λb, b ≠ 0
  • x = x + λa, y = y + λb, z = z + λc
  • (x − x)/a = (y − y)/b = (z − z)/c, for nonzero denominators

Angle between two lines

  • cos θ = |aa + bb + cc| / (LL)
  • L = √(a + b + c), L = √(a + b + c)
  • perpendicular: aa + bb + cc = 0
  • parallel directions: (a,b,c) = k(a,b,c), k ≠ 0

Shortest distance between lines

  • skew: d = |(b × b) · (a − a)| / |b × b|
  • parallel: d = |b × (a − a)| / |b|

Line of shortest distance

  • PQ · b = 0 and PQ · b = 0
  • d = |PQ| = |(b × b) · (a − a)| / |b × b|
  • r = p + t(b × b)

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