Introduction to Three Dimensional Geometry – NCERT Class 11 Mathematics Chapter 11 – Coordinate System, Points, Distances, and Basic Concepts

Introduces the three-dimensional coordinate system with mutually perpendicular axes and planes, defines coordinates of points in space, explains octants, derives the distance formula between points in 3D, discusses collinearity, planes and lines, and provides practical examples and exercises including centroid calculation. Also includes historical notes on the development of 3D geometry.

Updated: 8 months ago

Categories: NCERT, Class XI, Mathematics, Three Dimensional Geometry, Coordinate Geometry, Analytical Geometry, Chapter 11

Tags: Three Dimensional Geometry, 3D Coordinate System, Coordinates of Points, Octants, Distance Formula, Collinearity, Analytical Geometry, Cartesian Coordinates, NCERT Class 11, Mathematics, Chapter 11

Introduction to Three Dimensional Geometry: Class 11 NCERT Chapter 11 - Ultimate Study Guide, Notes, Questions, Quiz 2025

Full Chapter Summary & Detailed Notes - Three Dimensional Geometry Class 11 NCERT

Overview & Key Concepts

Chapter Goal: Extend 2D coordinates to 3D space for points, distances. Builds on 2D (Ch10). Exam Focus: Axes/planes, octants, distance formula. 2025 Updates: More apps like physics trajectories. Fun Fact: Euler systematized 3D coords (1748). Core Idea: Three perpendicular axes for space location. Real-World: Flight paths, room positioning. Ties: Vectors (Ch10), Lines (Ch12). Expanded: Full subtopics with explanations, visuals from PDF.
Wider Scope: From axes to distance in space (PDF covers coords, distance).
Expanded Content: Octants signs, collinearity, locus equations.

11.1 Introduction: From 2D to 3D

2D: Two axes for plane points. 3D: Need three for space (e.g., ball trajectory, bulb height). Coordinates: Perp distances from three planes (floor, walls). E.T. Bell quote: Math as queen/servant of sciences.

11.2 Coordinate Axes and Planes

Three mutually perp planes intersect at O: Axes XOX', YOY', ZOZ'. Planes: XY, YZ, ZX. Origin O; positive directions: Right (X), front (Y), up (Z). Octants: 8 parts (I: +++ to VIII: ---), like quadrants.

11.3 Coordinates of a Point

Point P(x,y,z): Drop perp to XY (M), then to X (L). x=OL, y=LM, z=MP. Signs per octant (Table 11.1). Alternative: Planes thru P parallel to coords meet axes at A(x,0,0), B(0,y,0), C(0,0,z). Origin (0,0,0); axes points (x,0,0) etc.

11.4 Distance between Two Points

P(x1,y1,z1), Q(x2,y2,z2): Form rectangular box, diagonal PQ. Pythagoras in 3D: $$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$. From origin: $$\sqrt{x^2 + y^2 + z^2}$$. Apps: Collinear (PQ + QR = PR), triangles.

Summary

3D Geometry: Axes/planes/octants for coords; distance extends Pythagoras. Master: Signs in octants, formula apps. Mantra: Three numbers for space position.

Why This Guide Stands Out

Visual octants table, step-by-step distance, free 2025 with MathJax.

Key Themes & Tips

Aspects: Coordinate system, distance calc, locus.
Tip: Memorize octant signs; verify collinear with distances.

Exam Case Studies

Aeroplane path distances; room bulb coords.

Project & Group Ideas

Model 3D axes with GeoGebra.
Apps: Trajectory simulation.

Key Definitions & Terms - Complete Glossary

All terms from chapter; detailed with examples, relevance. Expanded: 15+ terms with depth.

Coordinate Axes

Three mutually perp lines: X, Y, Z thru O. Relevance: Reference. Ex: XOX'. Depth: Positive directions defined.

Coordinate Planes

XY, YZ, ZX. Relevance: Divide space. Ex: XY as paper plane. Depth: Perp intersections.

Origin

Intersection O(0,0,0). Relevance: Zero point. Ex: All axes meet. Depth: Reference.

Octants

8 regions by planes. Relevance: Sign quadrants. Ex: I (+++). Depth: Roman numerals I-VIII.

Coordinates (x,y,z)

Perp distances from YZ, ZX, XY planes. Relevance: Point location. Ex: P(2,3,4). Depth: Ordered triplet.

X-Coordinate

Dist from YZ-plane. Relevance: Along X. Ex: Positive right. Depth: Signed.

Y-Coordinate

Dist from ZX-plane. Relevance: Along Y. Ex: Positive front. Depth: Signed.

Z-Coordinate

Dist from XY-plane. Relevance: Height. Ex: Positive up. Depth: Signed.

Rectangular Coordinate System

Perp axes in space. Relevance: Cartesian 3D. Ex: OXYZ. Depth: Extends 2D.

Distance Formula

$$ \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} $$. Relevance: Space dist. Ex: Between P,Q. Depth: Pythagoras 3D.

Collinear Points

Lie on straight line. Relevance: PQ + QR = PR. Ex: Verify distances. Depth: Vector alignment.

Locus

Set of points satisfying condition. Relevance: Equations. Ex: PA² + PB² = k. Depth: Sphere-like.

Centroid

Average coords: $$ \left( \frac{x_1+x_2+x_3}{3}, \dots \right) $$. Relevance: Triangle center. Ex: Find C given G,A,B. Depth: Balance point.

Perpendicular Planes

Mutually perp. Relevance: Axes def. Ex: XY perp Z. Depth: 90° angles.

Foot of Perpendicular

Projection point. Relevance: Coord calc. Ex: PM on XY. Depth: Shortest dist.

Ordered Triplet

(x,y,z) for point. Relevance: Unique ID. Ex: One-to-one. Depth: Real numbers.

Equidistant Locus

Points equal dist from A,B. Relevance: Perp bisector plane. Ex: Equation 10x + 6y -18z=0. Depth: Midplane.

Tip: Octant signs via table; distance always sqrt of sums. Depth: Properties like symmetry. Errors: Wrong signs. Historical: Euler 1748. Interlinks: Ch10 vectors. Advanced: Direction cosines. Real-Life: GPS coords. Graphs: Plot points in 3D. Coherent: Axes → Coords → Distance.

Additional: Parallelogram vectors. Pitfalls: Octant mix-up.

30 Questions & Answers - NCERT Based (Class 11) - From Exercises 11.1 & 11.2 Variations

Based on NCERT Ex 11.1 (coords/octants) + 11.2 (distance). Part A: 10 (1 mark short), Part B: 10 (4 marks medium), Part C: 10 (8 marks long). Answers point-wise, numerical stepwise with MathJax.

Part A: 1 Mark Questions (10 Qs - Short from Ex 11.1 & Variations)

1. Coordinates on x-axis?

1 Mark Answer:

$$ (x, 0, 0) $$

2. What divides space into octants?

1 Mark Answer:

Coordinate planes

3. Origin coordinates?

1 Mark Answer:

$$ (0,0,0) $$

4. Octant I signs?

1 Mark Answer:

+, +, +

5. YZ-plane points?

1 Mark Answer:

$$ (0, y, z) $$

6. Distance from origin formula?

1 Mark Answer:

$$ \sqrt{x^2 + y^2 + z^2} $$

7. XZ-plane y-coordinate?

1 Mark Answer:

8. Number of octants?

1 Mark Answer:

9. Collinear test?

1 Mark Answer:

PQ + QR = PR

10. X-coordinate from?

1 Mark Answer:

YZ-plane distance

Part B: 4 Marks Questions (10 Qs - Medium from Ex 11.2)

1. Distance (2,3,5) to (4,3,1)? (Ex 11.2 Q1 i)

4 Marks Answer (Step-by-Step):

Step 1: Δx=2, Δy=0, Δz=-4
Step 2: $$ \sqrt{4 + 0 + 16} = \sqrt{20} = 2\sqrt{5} $$
Relevance: Basic dist.

2. Octant for (-3,1,2)? (Ex 11.1 Q3)

4 Marks Answer (Step-by-Step):

Step 1: Signs: -, +, +
Step 2: Octant II
Relevance: Table 11.1.

3. Distance (-3,7,2) to (2,4,-1)? (Ex 11.2 Q1 ii)

4 Marks Answer (Step-by-Step):

Step 1: Δx=5, Δy=-3, Δz=-3
Step 2: $$ \sqrt{25 + 9 + 9} = \sqrt{43} $$
Relevance: Calc.

4. Collinear (-2,3,5),(1,2,3),(7,0,-1)? (Ex 11.2 Q2)

4 Marks Answer (Step-by-Step):

Step 1: PQ=$$ \sqrt{14} $$, QR=$$ \sqrt{56} $$, PR=$$ \sqrt{126} $$
Step 2: $$ \sqrt{14} + 2\sqrt{14} = 3\sqrt{14} = \sqrt{126} $$
Relevance: Sum equals.

5. Equidistant from (1,2,3),(3,2,-1)? (Ex 11.2 Q4)

4 Marks Answer (Step-by-Step):

Step 1: Set dist equal, square
Step 2: Simplify: 2x - 4z = 0 or x=2z
Relevance: Plane eq.

6. Sum dist from (4,0,0),(-4,0,0)=10? (Ex 11.2 Q5)

4 Marks Answer (Step-by-Step):

Step 1: $$ \sqrt{(x-4)^2 + y^2 + z^2} + \sqrt{(x+4)^2 + y^2 + z^2} = 10 $$
Step 2: Ellipse in x=0 plane? Wait, hyperbola-like.
Relevance: Locus.

7. Distance (-1,3,-4) to (1,-3,4)? (Ex 11.2 Q1 iii)

4 Marks Answer (Step-by-Step):

Step 1: Δx=2, Δy=-6, Δz=8
Step 2: $$ \sqrt{4 + 36 + 64} = \sqrt{104} = 2\sqrt{26} $$
Relevance: Symmetric.

8. Isosceles (0,7,-10),(1,6,-6),(4,9,-6)? (Ex 11.2 Q3 i)

4 Marks Answer (Step-by-Step):

Step 1: Calc AB=$$ \sqrt{11} $$, BC=5, AC=5
Step 2: BC=AC, isosceles
Relevance: Equal sides.

9. Right triangle (0,7,10),(-1,6,6),(-4,9,6)? (Ex 11.2 Q3 ii)

4 Marks Answer (Step-by-Step):

Step 1: AB=$$ \sqrt{2} $$, BC=$$ \sqrt{18} $$, AC=$$ \sqrt{20} $$
Step 2: AB² + BC² = AC²? 2+18=20 yes
Relevance: Pythagoras.

10. Parallelogram (-1,2,1),(1,-2,5),(4,-7,8),(2,-3,4)? (Ex 11.2 Q3 iii)

4 Marks Answer (Step-by-Step):

Step 1: Vectors AB=AD? Calc dist equal opposites
Step 2: Yes, parallelogram
Relevance: Opposite equal.