Last Updated: June 1, 2026

📋 Table of Contents
What is Three Dimensional Geometry Revision Notes?
Introduction
1. Coordinate Planes and Axes and 3D
2. Coordinates of a Point and Octants
3. Distance Formula and 3D
4. Section Formula
5. Centroid of a Triangle
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Three Dimensional Geometry Revision Notes?
Introduction
1. Coordinate Planes and Axes and 3D
2. Coordinates of a Point and Octants
3. Distance Formula and 3D
4. Section Formula
5. Centroid of a Triangle
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Three Dimensional Geometry Class 11 Physics Revision — JEE & NEET 2026 Grandmaster Guide

What is Three Dimensional Geometry Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Coordinate Planes: XY plane (z=0), YZ plane (x=0), ZX plane (y=0).

Octants: The space is divided into 8 octants y the three planes.

Distance Formula (P₁P₂): √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].

Section Formula (Internal): [(mx₂ + nx₁) / (m+n), (my₂ + ny₁) / (m+n), (mz₂ + nz₁) / (m+n)].

Centroid of a Triangle: [(x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3]. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

Three Dimensional Geometry extends the Cartesian coordinate system into space, providing the foundation for engineering, flight navigation, n Vector Algebra. Master the Eight Octants, the 3D distance formula, n section formulas to excel and spatial modeling and advanced physics. This class 11 Math Chapter 12 summary provides all essential formulas for JEE and Board exam success. The transition from 2D to 3D geometry is like going from a flat map to the real world.

1. Coordinate Planes and Axes and 3D

In three dimensions, we use three mutually perpendicular lines passing through the origin: the X, Y, n Z axes.

Coordinate Planes:
- XY Plane: Contains x and y axes. Equation: z = 0.
- YZ Plane: Contains y and z axes. Equation: x = 0.
- ZX Plane: Contains z and x axes. Equation: y = 0.

2. Coordinates of a Point and Octants

A point P and space is represented y (x, y, z). These axes divide the space into 8 octants.

Sign Convention for Octants:
- I: (+, +, +)
- II: (-, +, +)
- III: (-, -, +)
- IV: (+, -, +)
- V: (+, +, -)
- VI: (-, +, -)
- VII: (-, -, -)
- VIII: (+, -, -)

3. Distance Formula and 3D

The distance between two points P₁(x₁, y₁, z₁) n P₂(x₂, y₂, z₂) is given y: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] This formula is an extension of the Pythagorean theorem into three dimensions.

4. Section Formula

If a point R divides the line segment joining P(x₁, y₁, z₁) n Q(x₂, y₂, z₂) n the ratio m : n:

Internal Division: R = [(mx₂ + nx₁) / (m+n), (my₂ + ny₁) / (m+n), (mz₂ + nz₁) / (m+n)]
External Division: R = [(mx₂ - nx₁) / (m-n), (my₂ - ny₁) / (m-n), (mz₂ - nz₁) / (m-n)]
Midpoint: M = [(x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2]

5. Centroid of a Triangle

The centroid of a triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), n (x₃, y₃, z₃) is: G = [(x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3, (z₁ + z₃ + z₃) / 3]

Comprehensive Exam Strategy (Q&A)

Q1: Name the octant and which the point (-3, 1, -2) lies. Answer:

x is negative, y is positive, z is negative.
Sign pattern: (-, +, -).
This corresponds to Octant VI.

Q2: Find the distance between P(1, -3, 4) n Q(-4, 1, 2). Answer:

d = √[(-4 - 1)² + (1 - (-3))² + (2 - 4)²]
d = √[(-5)² + (4)² + (-2)²]
d = √[25 + 16 + 4] = √45
d = 3√5 units.

Q3: Find the coordinates of the point which divides the line joining (1, -2, 3) n (3, 4, -5) internally and the ratio 2 : 3. Answer:

x = [2(3) + 3(1)] / 5 = 9/5
y = [2(4) + 3(-2)] / 5 = 2/5
z = [2(-5) + 3(3)] / 5 = -1/5
Point = (9/5, 2/5, -1/5).

Related Revision Notes

Chapter 11: Conic Sections
Chapter 13: limits n Derivatives
[External Reference: NCERT Class 11 Math Chapter 12 (Authoritative Source)]

Conclusion

3D Geometry is the gateway to understanding spatial relationships. By mastering the octant signs and adapting the distance and section formulas from 2D to 3D, you build the mental framework required for advanced physics and engineering. Always visualize the point relative to the coordinate planes, n remember that x, y, or z being zero tells you exactly which plane you're on! Keep your spatial orientation clear.

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

📚 Related Topics

Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

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🪤 The 5 Mistakes That Cost Marks

Incorrect assumption of coordinate axes: Students often assume that the coordinate axes in 3D geometry are always aligned with the edges of the cube or the object being described, which can lead to incorrect calculations of direction cosines, projections, and distances.
Confusion between direction cosines and direction ratios: Direction cosines and direction ratios are often confused with each other, which can result in incorrect calculations, especially when dealing with the direction of lines and planes in 3D space.
Inadequate consideration of quadrants: In 3D geometry, it's essential to consider the quadrants in which the coordinates of a point lie, as it can affect the sign of the coordinates and the resulting calculations, especially when dealing with distance, direction, and orientation.
Misapplication of distance and section formulas: The distance and section formulas in 3D geometry have specific conditions and constraints that must be met before they can be applied, and misapplying these formulas can lead to incorrect results and loss of marks.
Ignoring the orientation of planes and lines: Failing to consider the orientation of planes and lines in 3D space can result in incorrect calculations, especially when dealing with angles between planes, perpendicular distances, and projections of points and lines onto planes and other lines.

🔁 Last 5 Minutes Box

Direction Cosines: \\cos α, \\cos β, \\cos γ are direction cosines of a line, where α, β, γ are angles made with x, y, z axes.
Direction Ratios: a, b, c are direction ratios of a line, related to direction cosines by a = λ\\cos α, b = λ\\cos β, c = λ\\cos γ.
Distance between two points: √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
Section Formula: point (x, y, z) divides line joining (x1, y1, z1) and (x2, y2, z2) in ratio m:n, then x = (mx2 + nx1)/(m+n), y = (my2 + ny1)/(m+n), z = (mz2 + nz1)/(m+n).
Midpoint: ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2).
Centroid of a triangle: ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3, (z1 + z2 + z3)/3).
Equation of a plane: ax + by + cz + d = 0, where a^2 + b^2 + c^2 ≠ 0.
Normal to a plane: (a, b, c).
Distance of a point (x1, y1, z1) from a plane: |ax1 + by1 + cz1 + d| / √(a^2 + b^2 + c^2).
Angle between two planes: \\cos θ = |a1a2 + b1b2 + c1c2| / (√(a1^2 + b1^2 + c1^2) * √(a2^2 + b2^2 + c2^2)).
Equation of a line: (x - x1)/a = (y - y1)/b = (z - z1)/c.
Shortest distance between two lines: |(x2 - x1)b1c2 - (y2 - y1)c1a2 + (z2 - z1)a1b2| / √((b1c2 - c1b2)^2 + (c1a2 - a1c2)^2 + (a1b2 - b1*a2)^2)

Last Updated: June 1, 2026

📋 Table of Contents
What is Three Dimensional Geometry Revision Notes?
Introduction
1. Coordinate Planes and Axes and 3D
2. Coordinates of a Point and Octants
3. Distance Formula and 3D
4. Section Formula
5. Centroid of a Triangle
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Three Dimensional Geometry Revision Notes?
Introduction
1. Coordinate Planes and Axes and 3D
2. Coordinates of a Point and Octants
3. Distance Formula and 3D
4. Section Formula
5. Centroid of a Triangle
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Three Dimensional Geometry Class 11 Physics Revision — JEE & NEET 2026 Grandmaster Guide

What is Three Dimensional Geometry Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Coordinate Planes: XY plane (z=0), YZ plane (x=0), ZX plane (y=0).

Octants: The space is divided into 8 octants y the three planes.

Distance Formula (P₁P₂): √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].

Section Formula (Internal): [(mx₂ + nx₁) / (m+n), (my₂ + ny₁) / (m+n), (mz₂ + nz₁) / (m+n)].

Centroid of a Triangle: [(x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3]. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

1. Coordinate Planes and Axes and 3D

In three dimensions, we use three mutually perpendicular lines passing through the origin: the X, Y, n Z axes.

Coordinate Planes:
- XY Plane: Contains x and y axes. Equation: z = 0.
- YZ Plane: Contains y and z axes. Equation: x = 0.
- ZX Plane: Contains z and x axes. Equation: y = 0.

2. Coordinates of a Point and Octants

A point P and space is represented y (x, y, z). These axes divide the space into 8 octants.

Sign Convention for Octants:
- I: (+, +, +)
- II: (-, +, +)
- III: (-, -, +)
- IV: (+, -, +)
- V: (+, +, -)
- VI: (-, +, -)
- VII: (-, -, -)
- VIII: (+, -, -)

3. Distance Formula and 3D

4. Section Formula

If a point R divides the line segment joining P(x₁, y₁, z₁) n Q(x₂, y₂, z₂) n the ratio m : n:

Internal Division: R = [(mx₂ + nx₁) / (m+n), (my₂ + ny₁) / (m+n), (mz₂ + nz₁) / (m+n)]
External Division: R = [(mx₂ - nx₁) / (m-n), (my₂ - ny₁) / (m-n), (mz₂ - nz₁) / (m-n)]
Midpoint: M = [(x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2]

5. Centroid of a Triangle

The centroid of a triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), n (x₃, y₃, z₃) is: G = [(x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3, (z₁ + z₃ + z₃) / 3]

Comprehensive Exam Strategy (Q&A)

Q1: Name the octant and which the point (-3, 1, -2) lies. Answer:

x is negative, y is positive, z is negative.
Sign pattern: (-, +, -).
This corresponds to Octant VI.

Q2: Find the distance between P(1, -3, 4) n Q(-4, 1, 2). Answer:

d = √[(-4 - 1)² + (1 - (-3))² + (2 - 4)²]
d = √[(-5)² + (4)² + (-2)²]
d = √[25 + 16 + 4] = √45
d = 3√5 units.

Q3: Find the coordinates of the point which divides the line joining (1, -2, 3) n (3, 4, -5) internally and the ratio 2 : 3. Answer:

x = [2(3) + 3(1)] / 5 = 9/5
y = [2(4) + 3(-2)] / 5 = 2/5
z = [2(-5) + 3(3)] / 5 = -1/5
Point = (9/5, 2/5, -1/5).

Related Revision Notes

Chapter 11: Conic Sections
Chapter 13: limits n Derivatives
[External Reference: NCERT Class 11 Math Chapter 12 (Authoritative Source)]

Conclusion

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

📚 Related Topics

Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

📚 Related Topics

Continue your revision with these related guides:

🪤 The 5 Mistakes That Cost Marks

Incorrect assumption of coordinate axes: Students often assume that the coordinate axes in 3D geometry are always aligned with the edges of the cube or the object being described, which can lead to incorrect calculations of direction cosines, projections, and distances.
Confusion between direction cosines and direction ratios: Direction cosines and direction ratios are often confused with each other, which can result in incorrect calculations, especially when dealing with the direction of lines and planes in 3D space.
Inadequate consideration of quadrants: In 3D geometry, it's essential to consider the quadrants in which the coordinates of a point lie, as it can affect the sign of the coordinates and the resulting calculations, especially when dealing with distance, direction, and orientation.
Misapplication of distance and section formulas: The distance and section formulas in 3D geometry have specific conditions and constraints that must be met before they can be applied, and misapplying these formulas can lead to incorrect results and loss of marks.
Ignoring the orientation of planes and lines: Failing to consider the orientation of planes and lines in 3D space can result in incorrect calculations, especially when dealing with angles between planes, perpendicular distances, and projections of points and lines onto planes and other lines.

🔁 Last 5 Minutes Box

Direction Cosines: \\cos α, \\cos β, \\cos γ are direction cosines of a line, where α, β, γ are angles made with x, y, z axes.
Direction Ratios: a, b, c are direction ratios of a line, related to direction cosines by a = λ\\cos α, b = λ\\cos β, c = λ\\cos γ.
Distance between two points: √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
Section Formula: point (x, y, z) divides line joining (x1, y1, z1) and (x2, y2, z2) in ratio m:n, then x = (mx2 + nx1)/(m+n), y = (my2 + ny1)/(m+n), z = (mz2 + nz1)/(m+n).
Midpoint: ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2).
Centroid of a triangle: ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3, (z1 + z2 + z3)/3).
Equation of a plane: ax + by + cz + d = 0, where a^2 + b^2 + c^2 ≠ 0.
Normal to a plane: (a, b, c).
Distance of a point (x1, y1, z1) from a plane: |ax1 + by1 + cz1 + d| / √(a^2 + b^2 + c^2).
Angle between two planes: \\cos θ = |a1a2 + b1b2 + c1c2| / (√(a1^2 + b1^2 + c1^2) * √(a2^2 + b2^2 + c2^2)).
Equation of a line: (x - x1)/a = (y - y1)/b = (z - z1)/c.
Shortest distance between two lines: |(x2 - x1)b1c2 - (y2 - y1)c1a2 + (z2 - z1)a1b2| / √((b1c2 - c1b2)^2 + (c1a2 - a1c2)^2 + (a1b2 - b1*a2)^2)