Three Dimensional Geometry Class 12 Notes

The Three Dimensional Geometry Class 12 Notes hosted here summarise Class 12 Mathematics Chapter 11 Three Dimensional Geometry in the order taught by the NCERT. Each result in the Three Dimensional Geometry Class 12 Notes is stated formally, then explained in a single short paragraph, then illustrated with one solved examples. The notes keep the notation of the PDF so cross-referencing remains direct.

Pages: 26 Core formulas: 22 Solved templates: 8 Syllabus: 2026-27

CBSE Boards: 6 to 8 marks in Class 12 Maths (the Vectors and 3D Geometry unit carries about 14 marks combined with Chapter 10)
JEE Main: 2 to 3 questions on direction cosines, equation of a line, plane equations, and skew-line distance (about 5 to 7 percent of the Maths paper)

Chapter 11 Three Dimensional Geometry Notes PDF

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Student Pulse - Three Dimensional Geometry Difficulty (March 2026 survey of 12,840 Class 12 students):

73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.

The lines-and-planes content sits across nine sections in the current NCERT edition. The Three Dimensional Geometry Class 12 Notes below sequence them as the Three Dimensional Geometry Class 12 Notes does, marking the four pieces that historically generate the long-answer questions: angle between two lines, shortest distance for skew lines, plane in normal form, and distance of a point from a plane.

These Three Dimensional Geometry Class 12 Notes are written by Collegedunia's Class 12 Maths editors and cross-checked against the 2026-27 NCERT textbook and the past six years of CBSE marking schemes, so the Three Dimensional Geometry Class 12 Notes you revise match what examiners actually award marks for.

Three Dimensional Geometry Notes - Class 12 Maths

Class 12 Maths Ncert Notes Chapter 11 Three Dimensional Geometry: Section Sequence

The 2026-27 NCERT presents Chapter 11 as a vectors-first treatment. After the opening on coordinate axes, the Three Dimensional Geometry Class 12 Notes introduces direction cosines, the vector and Cartesian forms of a line, the angle and distance results, and then the plane in four equivalent forms.

The walkthrough below preserves that order so your revision tracks the Three Dimensional Geometry Class 12 Notes page by page.

1. Direction cosines and direction ratios of a line

If a directed line L makes angles α, β, γ with the positive x-, y-, and z-axes, the cosines l = cosα, m = cosβ, n = cosγ are the direction cosines of L and satisfy: $$ l^2 + m^2 + n^2 = 1 $$

Any three numbers a, b, c proportional to l, m, n are direction ratios. To recover the cosines from the ratios: $$ l = \dfrac{a}{\sqrt{a^2+b^2+c^2}},\ \ m = \dfrac{b}{\sqrt{a^2+b^2+c^2}},\ \ n = \dfrac{c}{\sqrt{a^2+b^2+c^2}} $$

Quick Note: Direction cosines are unique up to sign, since a line has two senses. Direction ratios are unique only up to a non-zero scalar multiple, so (1, 2, 2) and (3, 6, 6) describe the same line.

2. Direction cosines of a line through two given points

For a line through P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) , the direction ratios are (x₂ - x₁, y₂ - y₁, z₂ - z₁) and: $$ l = \dfrac{x_2 - x_1}{PQ},\ m = \dfrac{y_2 - y_1}{PQ},\ n = \dfrac{z_2 - z_1}{PQ} $$ where PQ = √(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)² .

3. Vector and Cartesian equation of a line

A line through the point with position vector a⃗ and parallel to b⃗ is described by:

$$ \vec{r} = \vec{a} + \lambda \vec{b}, \quad \lambda \in \mathbb{R} $$

With a⃗ = x₁ î + y₁ ĵ + z₁ k̂ and b⃗ = a î + b ĵ + c k̂ , the symmetric Cartesian form is:

$$ \dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c} $$

Watch Out: If any of a, b, c is zero, do not divide by it. Write that coordinate as a separate equation. If b = 0, the line satisfies y = y₁ and the remaining two ratios stay equal.

4. Line through two given points

The line joining A(a⃗) and B(b⃗) has vector equation: $$ \vec{r} = \vec{a} + \lambda (\vec{b} - \vec{a}) $$ and Cartesian form: $$ \dfrac{x - x_1}{x_2 - x_1} = \dfrac{y - y_1}{y_2 - y_1} = \dfrac{z - z_1}{z_2 - z_1} $$

5. Angle between two lines

6. Shortest distance between two skew lines

7. Distance between two parallel lines

8. Equation of a plane (four equivalent forms)

9. Angle between two planes and between a line and a plane

10. Distance of a point from a plane

11. The three-step recipe for any Chapter 11 problem

Three Dimensional Geometry Video Walkthrough

Source: Magnet Brains on YouTube

Class 12 Maths Chapter 11 Three Dimensional Geometry: Important Topics for Boards and JEE

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

Concept checklist for the 2026-27 syllabus
Direction cosines and direction ratios	Identity l² + m² + n² = 1
DCs and DRs of a line through two points	Vector form r⃗ = a⃗ + λ b⃗
Cartesian symmetric form of a line	Line through two given points
Angle between two lines (vector and DR)	Parallel and perpendicular conditions
Shortest distance between skew lines	Distance between two parallel lines
Plane in vector and Cartesian general form	Plane in normal form r⃗ · n̂ = p
Plane in intercept form	Angle between two planes
Angle between a line and a plane	Distance of a point from a plane

Direction cosines of a line in 3D — symbols, identity, DR relation, sign convention

How Will Collegedunia's NCERT Class 12 Maths Chapter 11 Notes Help You?

Every formula is written with its physical reading and the configuration it applies to, so you never apply the skew-line formula to a parallel pair by accident.
The vector form and the Cartesian form of each result sit side by side, mirroring how CBSE alternates the question wording from year to year.
Plane-conversion rules (general to normal, general to intercept) are spelled out as procedures, not just identities, so the 1-mark sign-correction trap stops costing you marks.
The skew-line shortest-distance result is shown as both the vector triple product and the 3x3 determinant, matching the two phrasings CBSE has used since 2020.

Class 12 Maths Chapter 11 Formula Recall: Top Five Results

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

Result	Formula
Direction-cosine identity	l² + m² + n² = 1
Vector equation of a line	r⃗ = a⃗ + λ b⃗
Angle between two lines	cosθ = \|b₁⃗ · b₂⃗\|\|b₁⃗\|\|b₂⃗\|
Shortest distance, skew lines	d = \|(b₁⃗ × b₂⃗) · (a₂⃗ - a₁⃗)\|\|b₁⃗ × b₂⃗\|
Distance of point from plane	d = \|Ax₀ + By₀ + Cz₀ + D\|√A² + B² + C²

Class 12 Maths Chapter 11 CBSE Previous Year Question Trends

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

Year	Marks	Concept tag
2025	5	Shortest distance between two skew lines (vector form)
2025	3	Distance of a given point from a given plane
2024	5	Line through two points + shortest distance to a second line
2023	3	Direction cosines of the line joining two given points
2022	2	Angle between two given lines (Cartesian form)
2021	5	Shortest distance, Cartesian determinant form

Three Dimensional Geometry: Quick Revision Snapshot

Line equation in 3D: vector form versus cartesian form comparison

Class 12 Maths Chapter 11: Common Mistakes to Avoid

Confusing direction cosines (unique up to sign) with direction ratios (unique up to scalar). The identity l² + m² + n² = 1 holds for cosines, not ratios.
Dividing by zero in the Cartesian symmetric form when one of a, b, c is zero. Write that coordinate as a separate equation.
Writing cosθ = b₁⃗ · b₂⃗ / (|b₁⃗||b₂⃗|) without the modulus. CBSE wants the acute angle, so the modulus is mandatory.
Applying the skew-line formula to parallel lines. For parallel lines the cross product of direction vectors is zero and you must use |b⃗ × (a₂⃗ - a₁⃗)| / |b⃗| instead.
Converting Ax + By + Cz + D = 0 to normal form without correcting the sign so the right-hand side becomes non-negative. The unit normal must point away from the origin.
Treating "angle between line and plane" the same as "angle between two lines". The sine appears instead of the cosine because the angle is measured from the plane, not from the normal.

Class 12 Maths Chapter-wise CBSE Weightage Comparison

Ch 1 Relations and Functions

Ch 2 Inverse Trig Functions

Ch 3 Matrices

Ch 4 Determinants

Ch 5 Continuity and Differentiability

Ch 6 Application of Derivatives

Ch 7 Integrals

Ch 8 Application of Integrals

Ch 9 Differential Equations

Ch 10 Vector Algebra

Ch 11 Three Dimensional Geometry

Ch 12 Linear Programming

Ch 13 Probability

Related Resources for Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT Notes for Class 12 Maths: All Chapters

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

Chapter	NCERT Notes
Chapter 1	Relations and Functions Notes
Chapter 2	Inverse Trigonometric Functions Notes
Chapter 3	Matrices Notes
Chapter 4	Determinants Notes
Chapter 5	Continuity and Differentiability Notes
Chapter 6	Application of Derivatives Notes
Chapter 7	Integrals Notes
Chapter 8	Application of Integrals Notes
Chapter 9	Differential Equations Notes
Chapter 10	Vector Algebra Notes
Chapter 12	Linear Programming Notes
Chapter 13	Probability Notes