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.
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Introduction to Three Dimensional Geometry
Chapter 11: Mathematics - Ultimate Study Guide | NCERT Class 11 Notes, Questions, Examples & 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?
- $$ (x, 0, 0) $$
2. What divides space into octants?
- Coordinate planes
3. Origin coordinates?
- $$ (0,0,0) $$
4. Octant I signs?
- +, +, +
5. YZ-plane points?
- $$ (0, y, z) $$
6. Distance from origin formula?
- $$ \sqrt{x^2 + y^2 + z^2} $$
7. XZ-plane y-coordinate?
- 0
8. Number of octants?
- 8
9. Collinear test?
- PQ + QR = PR
10. X-coordinate from?
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- Step 1: Vectors AB=AD? Calc dist equal opposites
- Step 2: Yes, parallelogram
- Relevance: Opposite equal.
Part C: 8 Marks Questions (10 Qs - Long Detailed)
1. Full Ex 11.1 Q3: Octants for 8 points.
- (1,2,3): I; (4,-2,3): IV; (4,-2,-5): VII etc.
- Steps: Signs per table.
2. Ex 11.2 Q1 all: 4 distances.
- (i) $$ 2\sqrt{5} $$; (ii) $$ \sqrt{43} $$; (iii) $$ 2\sqrt{26} $$; (iv) 4
- Proof: Formula apply.
3. Collinear proof Ex 11.2 Q2 detailed.
- Step 1: Compute all dist
- Step 2: Verify sum
- Step 3: $$ \sqrt{14} + \sqrt{56} = \sqrt{70} + \sqrt{56} = wait, PQ+QR=PR $$
- Verify: Expand squares.
4. Equidistant locus Ex 11.2 Q4 full eq.
- Step 1: PA=PB, square both
- Step 2: Expand, simplify: x - z = 1? Wait, 2x - 4z = -2 or x=2z-1
- Step 3: Plane eq.
- Verify points.
5. Distance formula derivation.
- Step 1: Box with diags, right triangles PAQ, ANQ
- Step 2: PQ² = PA² + AQ² = PA² + AN² + NQ²
- Step 3: Deltas squared.
- Full: Pythagoras chain.
6. PA² + PB² = 2k² locus Ex 6.
- Step 1: Expand both, sum
- Step 2: 2x² + 2y² + 2z² -4x -14y +4z = 2k² -109
- Step 3: Sphere eq.
- Relevance: Midpoint plane.
7. Parallelogram Ex 7 detailed vectors.
- Step 1: AB=6, BC=$$ \sqrt{43} $$, CD=6, DA=$$ \sqrt{43} $$
- Step 2: Opposites equal
- Step 3: Diags AC=$$ \sqrt{3} $$, BD=$$ \sqrt{155} $$ unequal, not rect.
- Verify: Midpoint same.
8. Octants table explain signs.
- Step 1: Planes divide: X=0 YZ, etc.
- Step 2: +++ I, --+ V etc.
- Step 3: Proof: Directions.
- Full table.
9. Centroid Ex 9 find C.
- Step 1: G= avg: x=(3-1+x)/3=1 → x=-1? Wait, (3 + (-1) + x)/3=1 → x=-1
- Correct: For A(3,-5,7),B(-1,7,-6),G(1,1,1): x= (3-1+x)/3=1 →1+x= -1? Calc: 3-1=2, 2+x=3 →x=-1? PDF: x= (3 + (-1) + x)/3=1 →2+x=3,x=1
- Step 2: y=( -5+7 +y)/3=1 →2+y=3,y=1; z=(7-6+z)/3=1 →1+z=3,z=2
- C(1,1,2)
10. Misc Ex 1: Fourth vertex parallelogram.
- Step 1: D= A+B-C vector
- Step 2: (3-1 + (-1), -1+2+1, 2-4+2)=(1,2,0)
- Step 3: Verify opposites.
- Relevance: Vector add.
Tip: Distance formula key for 8 marks; octants quick recall.
Key Concepts - In-Depth Exploration
Core ideas with examples, pitfalls, interlinks.
3D Axes/Planes
Perp at O. Deriv: Extension 2D. Pitfall: Directions mix. Ex: Z up. Interlink: Ch10. Depth: Octants.
Coordinates Assignment
Perp drops. Deriv: Projections. Pitfall: Wrong plane. Ex: x from YZ. Interlink: Distance. Depth: Triplet bijection.
Octants & Signs
8 with sign patterns. Deriv: Plane divisions. Pitfall: Roman nums. Ex: II (-- + +? -++). Interlink: Quadrants. Depth: Table use.
Distance Formula
3D Pythagoras. Deriv: Box diagonal. Pitfall: Forget sqrt. Ex: Collinear check. Interlink: 2D dist. Depth: Origin special.
Collinearity
Dist sum. Deriv: Line property. Pitfall: Order. Ex: PQR on line. Interlink: Vectors. Depth: Section formula.
Locus Equations
Equidist sets. Deriv: Dist equal. Pitfall: Square both. Ex: Plane. Interlink: Ch12 lines. Depth: Sphere for sum sq.
Advanced: Direction ratios. Pitfalls: Sign errors. Interlinks: Ch12 direction. Real: 3D modeling. Depth: Parametric. Examples: Flight dist. Graphs: 3D plot. Errors: Axes labels. Tips: Visualize box for dist; table for octants.
Solved Examples - Book Examples with Simple Explanations
NCERT Examples 1-9 solved step-by-step.
Example 1: Coords of F if P(2,4,5) (Fig 11.3)
Simple Explanation: Y=0 for F.
- Step 1: Along OY=0
- Step 2: F(2,0,5)
- Simple Way: Projection.
Example 2: Octants (-3,1,2), (-3,1,-2)
Simple Explanation: Signs check.
- Step 1: -++ = II
- Step 2: -+ - = VI
- Simple Way: Table lookup.
Example 3: Dist P(1,-3,4) Q(-4,1,2)
Simple Explanation: Plug formula.
- Step 1: Δx=-5, Δy=4, Δz=-2
- Step 2: $$ \sqrt{25+16+4}= \sqrt{45}=3\sqrt{5} $$
- Simple Way: Differences squared.
Example 4: Collinear P(-2,3,5),Q(1,2,3),R(7,0,-1)
Simple Explanation: Sum dist.
- Step 1: PQ=$$ \sqrt{14} $$, QR=$$ 2\sqrt{14} $$, PR=$$ 3\sqrt{14} $$
- Step 2: Sum = total
- Simple Way: Squares: 14+56=70, but 126=9*14 wait, sqrt scale.
Example 5: Right triangle A(3,6,9),B(10,20,30),C(25,-41,5)?
Simple Explanation: Check Pythagoras.
- Step 1: AB²=686, BC²=4571, CA²=2709
- Step 2: No match sums
- Simple Way: Largest sq vs sum others.
Example 6: Locus PA² + PB² = 2k²
Simple Explanation: Expand sum.
- Step 1: Plug A(3,4,5),B(-1,3,-7)
- Step 2: 2x²+2y²+2z² -4x-14y+4z=2k²-109
- Simple Way: Collect terms.
Example 7: Parallelogram A(1,2,3),B(-1,-2,-1),C(2,3,2),D(4,7,6)
Simple Explanation: Sides equal.
- Step 1: AB=CD=6, BC=DA=$$ \sqrt{43} $$
- Step 2: Diags unequal
- Simple Way: Vector add.
Example 8: Equidistant A(3,4,-5),B(-2,1,4)
Simple Explanation: Set equal.
- Step 1: PA=PB square
- Step 2: 10x+6y-18z-29=0
- Simple Way: Subtract exps.
Example 9: Centroid (1,1,1), A(3,-5,7),B(-1,7,-6) find C
Simple Explanation: Average.
- Step 1: x=(3-1+x)/3=1 → x= -1? Calc: 2+x=3,x=1
- Step 2: y=1,z=2
- Simple Way: Solve for missing.
Interactive Quiz - Master 3D Geometry
10 MCQs with MathJax; 80%+ goal. Coords, distance.
Quick Revision Notes & Mnemonics
Concise notes, mnemonics.
Basics
- Axes: X right, Y front, Z up; Planes XY paper
- Coords: x-YZ dist, y-ZX, z-XY
- Mnemonic: "XYZ: Right Front Up, Dist YZ ZX XY" (RFU DYZ X)
Octants
- I +++ , II -++ , III --+ , IV + - +; V ++- etc.
- Table: Signs alternate
- Mnemonic: "Positive X Y Z First, Flip per plane" (PXYZ FFP)
Distance
- $$ \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2} $$
- Origin: $$ \sqrt{x^2+y^2+z^2} $$
- Mnemonic: "Delta Squared Sum Sqrt" (DSS S)
Special Points
- X-axis (x,0,0); YZ (0,y,z)
- Collinear: Sum dist
- Mnemonic: "Axis Zero Others, Sum Equals Total" (AZO SET)
Locus
- Equidist: Plane mid
- Sum sq: Sphere
- Mnemonic: "Equal Dist Plane, Sum Square Sphere" (EDP SSS)
Centroid
- Avg coords /3
- Solve for missing
- Mnemonic: "Three Average Divide Three" (TAD T)
Overall Mnemonic: "Axes Octants Coords Distance Locus" (AOCD L). Flashcards for signs.
Formulas & Notations - All Key
| Formula/Notation | Description | Example | Usage |
|---|---|---|---|
| $$ (x,y,z) $$ | Point coords | P(1,2,3) | Location |
| Origin: (0,0,0) | Intersection | All zero | Reference |
| X-axis: (x,0,0) | On axis | (5,0,0) | Special |
| $$ PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} $$ | Distance | Δ=1,1,1: $$ \sqrt{3} $$ | Space dist |
| From O: $$ \sqrt{x^2 + y^2 + z^2} $$ | Origin dist | (3,4,0):5 | Radius |
| Octant I: +++ | Signs | (1,1,1) | Region |
| Centroid: $$ \left( \frac{x_1+x_2+x_3}{3}, \dots \right) $$ | Triangle center | Avg | Balance |
| Equidist: PA=PB | Locus plane | Perp bisector | Eq derivation |
| PA² + PB² = k | Sphere locus | Expand sum | Mid sphere |
Tip: Memorize distance, octant table.
Derivations & Proofs - Solved Step-by-Step
Derivation 1: Distance Formula
Step-by-Step:
- Step 1: Box P Q, planes parallel coords
- Step 2: Right Δ PAQ: PQ²=PA² + AQ²
- Step 3: AQ²=AN² + NQ²
- Conclusion: Sum deltas². Proof: 3D Pythagoras.
Derivation 2: Coordinates Bijection
Step-by-Step:
- Step 1: Point → perps to planes/axes
- Step 2: Triplet (x,y,z)
- Step 3: Reverse: Axes points, parallel planes intersect P
- Conclusion: One-to-one. Proof: Unique perps.
Proof 1: Octant Signs
Step-by-Step:
Proof 2: Collinear Dist
Step-by-Step:
- Step 1: On line, segments add
- Step 2: Pythagoras not, but vector: |P-R| = |P-Q| + |Q-R| if aligned
- Conclusion: Sum equals. Proof: Triangle ineq equality.
Proof 3: Equidistant Plane
Step-by-Step:
Tip: Use box for dist, signs for octants. Practice: Locus derive.
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