Class 12 Maths Chapter 11: Three Dimensional Geometry

Three Dimensional Geometry (3D Geometry) extends coordinate geometry from two dimensions to three dimensions by introducing the x, y, and z coordinate system. This chapter focuses on representing points, direction ratios, direction cosines, equations of lines, shortest distance between lines, and angles between lines. It is one of the most important chapters in CBSE Board, JEE Main, and JEE Advanced, where questions are often formula-based as well as application-oriented.

At ALLEN, these revision notes simplify Three Dimensional Geometry through concise explanations, essential formulas, graphical interpretation, solved examples, and exam-oriented tips. Class 12 Maths Chapter 11 revision notes help students strengthen conceptual understanding and improve problem-solving speed for board and competitive examinations.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	4–5 Hours
Key Concepts Covered	Coordinate System in 3D, Distance Between Two Points, Section Formula, Direction Cosines, Angle Between Two Lines

2.0Chapter 11 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution
Practice Paper with Answers

Test your Knowledge

question 1 of 5

The coordinates of a point in three-dimensional space are represented as:

1.(x, y)

2.(x, y, z)

3.(x, z)

4.(y, z)

Must-Revise Concepts

Coordinate Geometry

Coordinate Geometry PYQ with Solutions

Direction Cosines

Direction Ratios

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JEE Maths

Class 12 Chapter 11

Class 12 Maths Chapter 11: Three Dimensional Geometry

Three Dimensional Geometry (3D Geometry) extends coordinate geometry from two dimensions to three dimensions by introducing the x, y, and z coordinate system. This chapter focuses on representing points, direction ratios, direction cosines, equations of lines, shortest distance between lines, and angles between lines. It is one of the most important chapters in CBSE Board, JEE Main, and JEE Advanced, where questions are often formula-based as well as application-oriented.

At ALLEN, these revision notes simplify Three Dimensional Geometry through concise explanations, essential formulas, graphical interpretation, solved examples, and exam-oriented tips. Class 12 Maths Chapter 11 revision notes help students strengthen conceptual understanding and improve problem-solving speed for board and competitive examinations.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	4–5 Hours
Key Concepts Covered	Coordinate System in 3D, Distance Between Two Points, Section Formula, Direction Cosines, Angle Between Two Lines

2.0Chapter 11 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution
Practice Paper with Answers

Test your Knowledge

question 1 of 5

The coordinates of a point in three-dimensional space are represented as:

1.(x, y)

2.(x, y, z)

3.(x, z)

4.(y, z)

Must-Revise Concepts

Coordinate Geometry

Coordinate Geometry PYQ with Solutions

Direction Cosines

Direction Ratios

Three Dimensional Geometry

Three-dimensional geometry encompasses the mathematical study of shapes and figures in 3D space, involving three coordinates: the x-coordinate, y-coordinate, and z-coordinate. These coordinates define the precise location of a point in three-dimensional space, requiring three parameters for accurate positioning. Three-dimensional geometry is crucial for exams like JEE, as it features prominently and includes various operations on points within a 3D plane. It covers fundamental concepts such as direction cosines and Direction Ratios of a line, as well as the Direction cosines of a line passing through two points.

1.0Introduction to Three Dimensional Geometry

Three-dimensional geometry, also known as 3D geometry, is a branch of mathematics that deals with the study of shapes and figures in three-dimensional space. In three-dimensional space, objects not only have length and width (like in two-dimensional geometry) but also depth or height, creating a three-dimensional aspect.

2.0Section Formula in 3D

The section formula in 3D geometry helps determine the coordinates of a point that divides a line segment between two specified points in a given ratio. It is an extension of the section formula used in 2D geometry.

Let's consider two points in three-dimensional space:

Point A with coordinates (x₁, y₁, z₁)
Point B with coordinates (x₂, y₂, z₂)

Now, we want to find the coordinates of a point P that divides the line segment AB in the ratio m: n. The section formula in 3D can be expressed as follows:

$P (x, y, z) = \frac{mx _{2} + nx _{1}}{m + n}, \frac{my _{2} + ny _{1}}{m + n}, \frac{mz _{2} + nz _{1}}{m + n}$

1.0Direction Cosines and Direction Ratios of a Line

Direction Cosines

If 𝛼, 𝛽, 𝛾 are the angles made by a line with 𝑥 −axis, 𝑦 −axis, 𝑧 −axis, respectively, then cos𝛼, cos𝛽 & cos𝛾 are called direction cosines of a line, denoted by ℓ, 𝑚 & 𝑛 respectively.

(i) Direction cosines of a line

Take a vector $A = a \hat{i} + b \hat{j} + c \hat{k}$ parallel to a line whose 𝐷. 𝐶’𝑠 are to be found out.

$A . \hat{i} = a ∣ A ∣ cos α = a$

$cos α = \frac{a}{∣ A ∣} similarly, cos β = \frac{b}{∣ A ∣}; cos γ = \frac{c}{∣ A ∣}$

⇒ cos² α + cos² β + cos² γ = 1 ⇒ l² + m² + n² =1

$a \hat{i} + b \hat{j} + c \hat{k}$

(ii) Direction cosines of axes

Since the positive 𝑥 −axes makes angle 0°, 90°, 90° with axes of 𝑥, 𝑦 and 𝑧 respectively,

∴ 𝐷. 𝐶. ’𝑠 of 𝑥 − axis are 1, 0, 0.

𝐷. 𝐶. ’𝑠 of 𝑦 − axis are 0, 1, 0.

𝐷. 𝐶. ’𝑠 of 𝑧 − axis are 0, 0, 1.

Direction Ratios

Any three numbers 𝑎, 𝑏, 𝑐, proportional to direction cosines ℓ, 𝑚, 𝑛 are called direction ratios of the line. i.e. $\frac{1}{a} = \frac{m}{b} = \frac{n}{c}$ .

There can be infinitely many sets of direction ratios for a given line.

2.0Relation Between Direction Cosines and Direction Ratios:

$\frac{l}{a} = \frac{m}{b} = \frac{n}{c} ∴ \frac{l ^{2}}{a ^{2}} = \frac{m ^{2}}{b ^{2}} = \frac{n ^{2}}{c ^{2}} = \frac{l ^{2} + m ^{2} + n ^{2}}{a ^{2} + b ^{2} + c ^{2}} ∴ l = \frac{\pm a}{a ^{2} + b ^{2} + c ^{2}}; m = \frac{\pm b}{a ^{2} + b ^{2} + c ^{2}}; n = \frac{\pm c}{a ^{2} + b ^{2} + c ^{2}}$

3.0Direction Ratios and Direction Cosines of the Line Joining Two Points

Let us assume 𝐴 (𝑥₁, 𝑦₁, 𝑧₁) and 𝐵(𝑥₂, 𝑦₂, 𝑧₂) be two points, then Direction Ratios of AB are 𝑥₂– 𝑥₁, 𝑦₂– 𝑦₁, 𝑧₂ – 𝑧₁ and the Direction Cosines of 𝐴𝐵 are

$\frac{l}{r} (x_{2} - x_{1}), \frac{1}{r} (y_{2} - y_{1}), \frac{1}{r} (z_{2} - z_{1})$

where $r = [\sum (x_{2} - x_{1})^{2}]$

4.0Projection Of Line Segment Joining Two Points On Another Line

Consider A (x₁, y₁, z₁) and B (x₂, y₂, z₂).

The Projection of line segment AB onto a line with direction cosines l, m, n is given as:

l (x₂ − x₁) + m(y₂ − y₁) + n(z₂ − z₁)

5.0Angle Between Two Lines in 3 Dimensional Space

Angle between two lines in Three Dimensional Space whose direction cosines are (l₁, m₁, n₁) and (l₂, m₂, n). Then angle between them is

θ = cos^–1 (l₁l₂ + m₁m₂ + n₁n₂)

6.0Equation Of A Line In Space

A line is uniquely determined in three-dimensional space if:

(i) It passes through a given point and has a given direction vector, or

(ii) It passes through two distinct given points.

Equation of a line passing through a given point and parallel to a given vector b.

Let a be the position vector of a given point A w.r.t origin O and l be the line that passes through point A and is parallel to a given vector b. Let r be the position vector of an arbitrary point P on the line.

So, the equation of a line in a Vector Form is given by: r = a + λ b

If l, m, and n represent the direction cosines of the line, the equation of the line in Cartesian coordinates can be expressed as:

$\frac{x - x _{1}}{l} = \frac{y - y _{1}}{m} = \frac{z - z _{1}}{n}$

7.0Angle between Two Lines

Let 𝜃 represent the angle between the lines with direction cosines ℓ₁, 𝑚₁, 𝑛₁ and ℓ₂, 𝑚₂, 𝑛₂. Then, the cosine of 𝜃 is given by 𝜃 = ℓ₁ℓ₂ + 𝑚₁𝑚₂ + 𝑛₁𝑛₂. If direction ratios are 𝑎₁, 𝑏₁, 𝑐₁ and 𝑎₂, 𝑏₂, 𝑐₂ of two lines, then the angle 𝜃 between them is given by:

$cos θ = \frac{( a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} )}{( a _{1}^{2} + b _{1}^{2} + c _{1}^{2} ) ( a _{2}^{1} + b _{2}^{2} + c _{2}^{2} )}$

8.0Angle Between a Line and a Plane

Let equations of the line and plane be $\frac{x - x _{1}}{ℓ} = \frac{y - y _{1}}{m} = \frac{z - z _{1}}{n}$ and 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0, respectively, and 𝜃 be the angle which the line makes with the plane. Then

$(\frac{π}{2} - θ)$

is the angle between the line and the normal to the plane.

So, $sin θ = \frac{a l + bm + c n}{( a ^{2} + b ^{2} + c ^{2} ) ( l ^{2} + m ^{2} + n ^{2} )}$

Line is parallel to the plane

If 𝜃 = 0, i.e. if 𝑎ℓ + 𝑏𝑚 + 𝑐𝑛 = 0.

Line is perpendicular to the plane-

If the line is parallel to the normal of the plane i.e., if $\frac{a}{l} = \frac{b}{m} = \frac{c}{n}$

9.0Shortest Distance Between Two Lines

If two lines in space intersect at a common point, then their shortest distance between them is zero. Also, if two lines are parallel to each other, then the shortest distance between them will be the perpendicular distance, i.e. the length of the perpendicular drawn from a point on one line onto the other line.

10.0Distance Between Two Skew Lines

The length of the line intercepted between two lines, which is perpendicular to both the lines, is the shortest distance between them. Let the equations of the lines 𝐴𝐵 and 𝐶𝐷 be:

$\frac{x - x _{1}}{a _{1}} = \frac{y - y _{1}}{b _{1}} = \frac{z - z _{1}}{c _{1}}$

And $\frac{x - x _{2}}{a _{2}} = \frac{y - y _{2}}{b _{2}} = \frac{z - z _{2}}{c _{2}}$

$\frac{x _{2} - x _{1} a _{1} a _{2} y _{2} - y _{1} b _{1} b _{2} z _{2} - z _{1} c _{1} c _{2}}{( b _{1} c _{2} - b _{2} c _{1} ) ^{2} + ( c _{1} a _{2} - c _{2} a _{1} ) ^{2} + ( a _{1} b _{2} - a _{2} b _{1} ) ^{2}}$

11.0Three Dimensional Geometry Solved Examples

Example 1: A line OP makes with the x −axis an angle of measure 120° and with y −axis an angle of measure 60°. Find the angle made by the line with the z −axis.

Solution: 𝛼 = 120° and 𝛽 = 60°

∴ cos𝛼 = cos120° = $- \frac{1}{2}$ and cos𝛽 = cos 60° = $\frac{1}{2}$

but cos²𝛼 + cos² 𝛽 + cos²𝛾 = 1

$∴ (\frac{- 1}{2})^{2} + (\frac{1}{2})^{2} + cos^{2} γ = 1$

$\Rightarrow cos^{2} γ = 1 - \frac{1}{4} - \frac{1}{4} = \frac{1}{2}$

$\Rightarrow cos γ = \pm \frac{1}{2}$

∴ 𝛾 = 45° or 135°

Example 2: Determine the length of the projection of the line segment that connects the points (-1, 0, 3) and (2, 5, 1) onto the line with direction ratios 6, 2, 3.

Solution: The direction cosines ℓ, 𝑚, 𝑛 of the line are given by

The direction cosines l m , n of the line are given by

$\frac{l}{6} = \frac{m}{2} = \frac{n}{3} = \frac{ℓ ^{2} + m ^{2} + n ^{2}}{6 ^{2} + 2 ^{2} + 3 ^{2}} = \frac{1}{49} = \frac{1}{7}$

∴ $ℓ = \frac{6}{7}, m = \frac{2}{7}, n = \frac{3}{7}$

The required length of projection is given by

$= ∣ ℓ (x_{2} - x_{1}) + m (y_{2} - y_{1}) + n (z_{2} - z_{1}) ∣$

$= \frac{6}{7} [2 - (- 1) + \frac{2}{7} (5 - 0) + \frac{3}{7} (1 - 3)]$

$= \frac{6}{7} \times 3 + \frac{2}{7} \times 5 + \frac{3}{7} \times - 2 = \frac{18}{7} + \frac{10}{7} - \frac{6}{7} = \frac{18 + 10 - 6}{7} = \frac{22}{7}$

Example 3. If a line makes an angle 90°, 60°, and 30° with the positive direction of the x, y, and z-axis, respectively, find its direction cosines.

Solution: Let the d. c. 's of the lines be l, m, n. Then l = cos 90° = 0, m = cos 60° = $\frac{1}{2}$ , n = $cos 3 0^{\circ}$ $= \frac{3}{2}$

Example 4. Calculate the shortest distance between the lines l₁ and l₂, whose vector equations are

$r = \hat{i} + \hat{j} + λ (2 \hat{i} - \hat{j} + \hat{k})$ ….(1)

$r = 2 \hat{i} + \hat{j} - \hat{k} + μ (3 \hat{i} - 5 \hat{j} + 2 \hat{k})$ ….(2)

Solution: Comparing (1) and (2) with

$r = a_{1} + λ b_{1} and r = a_{2} + μ b_{2}$ respectively.

We get

$a_{1} = \hat{i} + \hat{j}, b_{1} = 2 \hat{i} - \hat{j} + \hat{k}$

$a_{2} = 2 \hat{i} + \hat{j} - \hat{k} and b_{2} = 3 \hat{i} - 5 \hat{j} + 2 \hat{k}$

Therefore $a_{2} - a_{1} = \hat{i} - \hat{k}$

and $b_{1} \times b_{2} = (2 \hat{i} - \hat{j} + \hat{k}) \times (3 \hat{i} - 5 \hat{j} + 2 \hat{k})$

$\hat{i} 23 \hat{j} - 1 - 5 \hat{k} 12 = 3 \hat{i} - \hat{j} - 7 \hat{k}$

So $b_{1} \times b_{2} = 9 + 1 + 49 = 59$

Hence , the shortest distance between the given lines is given by

$d = \frac{( b _{1} \times b _{2} ) \cdot ( a _{2} - a _{1} )}{b _{1} \times b _{1}} = \frac{∣3 - 0 + 7∣}{59} = \frac{10}{59}$

Example 4: How do you calculate distances in 3D space?

Solution: The distance between two points represented as (x₁, y₁, z₁) & (x₂, y₂, z₂) in 3D space can be found using the distance formula derived from the Pythagorean theorem:

d = (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

Example 5: How do you find the angle between two vectors in 3D space?

Solution: The angle (θ) between two vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) in 3D space can be calculated using the dot product formula: $cos θ = \frac{A \cdot B}{∣ A ∣∣ B ∣}$

Also Read:

3D Geometry	Coplanar Lines	Coordinate Geometry
Analytical Geometry	Distance Between Two Points	Statistics
Tangent and Normal	Angle Between Two Lines	Equation and shortest distance of skew lines

1.0Learning Outcomes

After completing this chapter, you will be able to:

Understand the three-dimensional coordinate system.
Calculate direction ratios and direction cosines.
Write equations of lines in vector and Cartesian forms.
Find the angle between two lines.
Determine the shortest distance between skew lines.
Solve geometric problems involving points and lines in space.
Solve JEE Main, JEE Advanced, and CBSE Board questions confidently.

2.05-Minute Quick Revision

Must-Revise Concepts

Coordinate System in 3D
Distance Between Two Points
Section Formula
Direction Ratios (DRs)
Direction Cosines (DCs)
Equation of a Line
Angle Between Two Lines
Coplanarity of Lines
Shortest Distance Between Two Lines
Symmetric Form of Line

Important Formula Recall

Distance Formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$
Direction Cosines: $l^{2} + m^{2} + n^{2} = 1$
Equation of Line: $\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c}$
Angle Between Two Lines: $cos θ = \frac{a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2}}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$

Important Properties / Shortcut Tips

Convert direction ratios into direction cosines whenever required.
Remember that direction cosines always satisfy (l^2+m^2+n^2=1).
Use vector notation to simplify line equations.
Check whether lines are parallel or perpendicular before lengthy calculations.
Memorize standard formulas for angles and distances.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Direction Ratios & Direction Cosines	High	Easy–Moderate
Equation of a Line	Very High	Moderate
Angle Between Two Lines	Very High	Moderate
Shortest Distance Between Two Lines	Very High	Moderate–High
Coplanarity of Lines	High	Moderate
Distance Between Point & Line	High	Moderate
Mixed 3D Geometry Problems	Very High	High

4.0Formula Handbook

Available soon

5.0Common Mistakes & JEE Tips

Common Mistake	Correct Approach
Confusing direction ratios with direction cosines	Direction cosines are normalized values of direction ratios.
Forgetting to normalize vectors	Convert DRs into DCs whenever required.
Using incorrect line equations	Identify the correct point and direction vector before forming the equation.
Sign errors in distance calculations	Apply the distance formula carefully.
Incorrect angle calculation	Use the dot product formula accurately.
Missing vector notation	Write equations systematically in vector or Cartesian form.
Algebraic simplification errors	Simplify expressions before substituting values.

6.0ALLEN Faculty Tips

Memorize all standard 3D Geometry formulas.
Practice converting direction ratios into direction cosines.
Revise vector algebra alongside this chapter.
Solve angle and shortest-distance problems regularly.
Draw rough diagrams to visualize spatial relationships.
Practice Previous Years' Questions (PYQs) for speed and accuracy.

7.0PYQ Trend Analysis

Three Dimensional Geometry is one of the highest-scoring chapters in JEE Main, JEE Advanced, and CBSE Board examinations. Most questions are formula-based and involve direction ratios, equations of lines, angles, and shortest distances.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Direction Ratios & Cosines	1	Easy
Equation of Line	1–2	Moderate
Angle Between Two Lines	1	Moderate
Shortest Distance	1–2	Moderate–High
Mixed 3D Problems	1	Moderate–High

8.0Most Common Question Types

Finding direction ratios and direction cosines.
Writing equations of lines in different forms.
Finding the angle between two lines.
Calculating the shortest distance between skew lines.
Determining the distance between a point and a line.
Solving coplanarity-based problems.
Applying vector methods in three-dimensional geometry.
Solving mixed conceptual and numerical questions.

9.0Smart Revision Strategy

Revise vector algebra before studying this chapter.
Memorize all standard 3D Geometry formulas.
Practice line equation problems.
Solve direction ratio and direction cosine questions.
Revise angle and distance formulas.
Practice shortest-distance problems.
Solve Previous Years' Questions (PYQs).
Finish revision with a timed chapter test.

Class 12 Maths Chapters List:- JEE Revision Notes

Chapter 1 Relations and Functions

Chapter 2 Inverse Trigonometric Functions

Chapter 3 Matrices

Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 Application of Derivatives

Chapter 7 Integrals

Chapter 8 Application of Integrals

Chapter 9 Differential Equations

Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 Probability

Frequently Asked Questions

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I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Three Dimensional Geometry

Three-dimensional geometry encompasses the mathematical study of shapes and figures in 3D space, involving three coordinates: the x-coordinate, y-coordinate, and z-coordinate. These coordinates define the precise location of a point in three-dimensional space, requiring three parameters for accurate positioning. Three-dimensional geometry is crucial for exams like JEE, as it features prominently and includes various operations on points within a 3D plane. It covers fundamental concepts such as direction cosines and Direction Ratios of a line, as well as the Direction cosines of a line passing through two points.

1.0Introduction to Three Dimensional Geometry

Three-dimensional geometry, also known as 3D geometry, is a branch of mathematics that deals with the study of shapes and figures in three-dimensional space. In three-dimensional space, objects not only have length and width (like in two-dimensional geometry) but also depth or height, creating a three-dimensional aspect.

2.0Section Formula in 3D

The section formula in 3D geometry helps determine the coordinates of a point that divides a line segment between two specified points in a given ratio. It is an extension of the section formula used in 2D geometry.

Let's consider two points in three-dimensional space:

Point A with coordinates (x₁, y₁, z₁)
Point B with coordinates (x₂, y₂, z₂)