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?
2. What divides space into octants?
6. Distance from origin formula?
1 Mark Answer:
- $$ \sqrt{x^2 + y^2 + z^2} $$
7. XZ-plane y-coordinate?
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.
Part C: 8 Marks Questions (10 Qs - Long Detailed)
1. Full Ex 11.1 Q3: Octants for 8 points.
8 Marks Answer (Step-by-Step Numerical):
- (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.
8 Marks Answer (Step-by-Step Numerical):
- (i) $$ 2\sqrt{5} $$; (ii) $$ \sqrt{43} $$; (iii) $$ 2\sqrt{26} $$; (iv) 4
- Proof: Formula apply.
3. Collinear proof Ex 11.2 Q2 detailed.
8 Marks Answer (Step-by-Step Numerical):
- 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.
8 Marks Answer (Step-by-Step Numerical):
- 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.
8 Marks Answer (Step-by-Step Numerical):
- 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.
8 Marks Answer (Step-by-Step Numerical):
- 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.
8 Marks Answer (Step-by-Step Numerical):
- 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.
8 Marks Answer (Step-by-Step Numerical):
- 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.
8 Marks Answer (Step-by-Step Numerical):
- 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.
8 Marks Answer (Step-by-Step Numerical):
- 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.
