Introduction to Three Dimensional Geometry — Mathematics Class 11 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Introduction to Three Dimensional Geometry (Class 11) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Introduction to Three Dimensional Geometry (CBSE & HBSE)
Extends Cartesian plane to space: coordinate triple (x, y, z), distance, and section formula in 3D. CBSE expects 4–5 marks; HBSE 3–4 marks. Short, formula-heavy.
Coordinate System & Octants
Three Mutually Perpendicular Axes
Three axes — x, y, z — meet at origin O at right angles, forming three coordinate planes:
| Plane | Equation |
|---|---|
| xy-plane | z = 0 |
| yz-plane | x = 0 |
| zx-plane | y = 0 |
A point P in space is uniquely represented by an ordered triple (x, y, z).
Octants
Three planes divide space into 8 octants (vs 4 quadrants in 2D):
| Octant | x | y | z |
|---|---|---|---|
| I | + | + | + |
| II | − | + | + |
| III | − | − | + |
| IV | + | − | + |
| V | + | + | − |
| VI | − | + | − |
| VII | − | − | − |
| VIII | + | − | − |
Coordinates of Special Points
- Origin: O(0, 0, 0).
- A point on x-axis: (x, 0, 0).
- A point on y-axis: (0, y, 0).
- A point on z-axis: (0, 0, z).
- A point on xy-plane: (x, y, 0).
Coordinate Axes vs Coordinate Planes
- Two perpendicular planes meet in an axis (e.g. xy ∩ xz = x-axis).
- All three planes meet at the origin.
Distance from Coordinate Planes
From point P(x, y, z):
- Distance from xy-plane = |z|
- Distance from yz-plane = |x|
- Distance from zx-plane = |y|
Distance from Origin
OP = √(x² + y² + z²).
Right-Hand Rule
The positive directions follow the right-hand rule: thumb → x, index → y, middle → z. This is the standard convention.
CBSE Tip: Always state which octant a point belongs to, identifying signs of x, y, z separately — that locks in the answer.
Distance Formula in Three Dimensions
Distance Between Two Points
For points A(x₁, y₁, z₁) and B(x₂, y₂, z₂): AB = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
Derived by applying Pythagoras twice — once in xy-plane and once vertically.
Distance Properties
- AB ≥ 0; equality iff A = B.
- AB = BA (symmetric).
- AB + BC ≥ AC (triangle inequality).
Collinearity Test (3D)
Three points A, B, C are collinear iff the sum of two of the three pairwise distances equals the third (longest):
- AB + BC = AC (if B is between A and C), or any cyclic version.
Equidistant Points
If P is equidistant from A and B in 3D, P lies on the perpendicular bisector plane of segment AB.
Sample Computations
- Distance from (1, 2, 3) to (4, 6, 3) = √(9 + 16 + 0) = 5.
- A point equidistant from A(1, 0, 0), B(0, 1, 0), C(0, 0, 1) and at distance d from origin satisfies x = y = z = d/√3.
Type of Triangle (3D)
Given three points, compute the three pairwise distances:
- All equal ⇒ equilateral.
- Two equal ⇒ isosceles.
- One squared = sum of other two squared ⇒ right-angled.
Useful in Class 12
All Class 12 vector and 3D geometry chapters use this formula extensively. Master it thoroughly here.
HBSE Tip: Always verify your answer by computing AB and BA — they must match exactly.
Section Formula in 3D
Internal Division
A point R(x, y, z) divides the segment joining A(x₁, y₁, z₁) and B(x₂, y₂, z₂) internally in the ratio m : n: x = (mx₂ + nx₁)/(m + n) y = (my₂ + ny₁)/(m + n) z = (mz₂ + nz₁)/(m + n)
External Division
For external division in ratio m : n (replace n by −n in formulas): x = (mx₂ − nx₁)/(m − n) y = (my₂ − ny₁)/(m − n) z = (mz₂ − nz₁)/(m − n)
Mid-Point Formula
Mid-point = section in ratio 1 : 1. Mid-point of A(x₁, y₁, z₁) and B(x₂, y₂, z₂): ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2).
Centroid of a Triangle (3D)
If vertices are (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃): Centroid G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3).
Centroid Divides the Median in 2 : 1
Median from vertex A to mid-point of opposite side BC is divided by the centroid G in the ratio 2 : 1 (from vertex).
Three Collinear Points and Section
For three collinear points A, B, C in 3D, one of them divides the other two in some ratio. To find the ratio, solve the section formula with the known coordinates.
Useful in Word Problems
- Find the point dividing the joining line of two cities in 3D (e.g. for transportation problems).
- Find centroid of a triangular region in space.
- Locate balance point of three masses in space.
Plane Through Three Points (Preview — Class 12)
Three non-collinear points determine a unique plane. (Detailed in Class 12.)
CBSE Tip: When asked for ratio of division, set unknown ratio as k : 1 and solve from the equation for any one coordinate; verify with the other two coordinates.
Frequently asked questions
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Do these notes follow CBSE and HBSE?
Yes. The Introduction to Three Dimensional Geometry notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Introduction to Three Dimensional Geometry chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Introduction to Three Dimensional Geometry.