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Introduction to Three Dimensional Geometry – Class 11 Maths Chapter 11 Notes & NCERT PDF

Class 11MathematicsChapter 11NCERT book: Mathematics - Class 11

Notes and a simple summary of Chapter 11 of the NCERT Class 11 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 11: Introduction to Three Dimensional Geometry

Introduction

Chapter 11, Introduction to Three Dimensional Geometry, extends the coordinate idea from a plane to space. A point in a plane needs two numbers; a flying bird, a hanging lamp or an aeroplane needs three. The chapter sets up three mutually perpendicular axes, names the planes and octants they create, and derives the distance formula in space. (In the rationalised syllabus the chapter covers sections 11.1–11.4 plus miscellaneous examples.)

Key Concepts & Theorems

11.2 Coordinate Axes and Coordinate Planes

  • Three mutually perpendicular lines XOX′, YOY′ and ZOZ′ meeting at the origin O are the x-, y- and z-axes of a rectangular Cartesian coordinate system.
  • Each pair of axes determines a coordinate plane: the XY-plane, YZ-plane and ZX-plane.
  • The three coordinate planes divide space into eight octants, named XOYZ, X′OYZ, X′OY′Z, XOY′Z, XOYZ′, X′OYZ′, X′OY′Z′ and XOY′Z′ and numbered I to VIII.

    • 11.3 Coordinates of a Point in Space

    From a point P drop perpendiculars to the three coordinate planes. The coordinates (x, y, z) of P are its distances from the YZ-, ZX- and XY-planes respectively, with signs.

  • Origin: (0, 0, 0).
  • Any point on the x-axis: (x, 0, 0); on the y-axis: (0, y, 0); on the z-axis: (0, 0, z).
  • Any point in the XY-plane: (x, y, 0); in the YZ-plane: (0, y, z); in the ZX-plane: (x, 0, z).

    • Signs in the octants (Table 11.1):

  • I (+, +, +), II (−, +, +), III (−, −, +), IV (+, −, +)
  • V (+, +, −), VI (−, +, −), VII (−, −, −), VIII (+, −, −)

    • 11.4 Distance between Two Points

    For P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):

    PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]

    The distance of P(x, y, z) from the origin is √(x² + y² + z²). This formula lets us test collinearity (PQ + QR = PR), classify triangles (isosceles, right-angled) and identify parallelograms.

    Worked Examples

    Example 1. If P is (2, 4, 5) in the book's figure, the point F lies in the ZX-plane below it, so its y-coordinate is 0: F = (2, 0, 5).

    Example 2 (octants). (−3, 1, 2) has signs (−, +, +) → octant II; (−3, 1, −2) has signs (−, +, −) → octant VI.

    Example 3 (distance). P(1, −3, 4), Q(−4, 1, 2): PQ = √(25 + 16 + 4) = √45 = 3√5.

    Example 4 (collinearity). P(−2, 3, 5), Q(1, 2, 3), R(7, 0, −1): PQ = √14, QR = √56 = 2√14, PR = √126 = 3√14. Since PQ + QR = PR, the points are collinear.

    Example 5 (right triangle test). A(3, 6, 9), B(10, 20, 30), C(25, −41, 5): AB² = 686, BC² = 4571, CA² = 2709. No sum of two squares equals the third (686 + 2709 = 3395 ≠ 4571), so ABC is not a right-angled triangle.

    Example 6 (locus). Points P with PA² + PB² = 2k², A(3, 4, 5), B(−1, 3, −7): expanding gives 2x² + 2y² + 2z² − 4x − 14y + 4z = 2k² − 109.

    Example 9 (centroid). The centroid of triangle ABC is (1, 1, 1), A = (3, −5, 7) and B = (−1, 7, −6). Using (x₁ + x₂ + x₃)/3 = 1 etc., C = (1, 1, 2).

    Formula Sheet

  • Point on x-axis (x, 0, 0); y-axis (0, y, 0); z-axis (0, 0, z)
  • Point in XY-plane (x, y, 0); YZ-plane (0, y, z); ZX-plane (x, 0, z)
  • PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
  • Distance from origin = √(x² + y² + z²)
  • Collinear P, Q, R ⇔ one distance equals the sum of the other two

    • Exam Focus

    Exercise 11.1 asks for coordinates of points on axes/planes and octants of given points; Exercise 11.2 asks for distances, collinearity, isosceles/right triangles, parallelograms and loci (e.g. points equidistant from (1, 2, 3) and (3, 2, −1), which simplifies to x − 2z = 0).

    Outline — "Show that P, Q, R are collinear": compute PQ, QR, PR in simplest surd form; show the largest equals the sum of the other two.

    Outline — "Verify a right-angled triangle": compute the three squared lengths; check whether one equals the sum of the other two (Pythagoras).

    Outline — "Name the octant": write the sign pattern of (x, y, z) and match it to Table 11.1.

    Key Terms

  • Three-dimensional space — त्रिविमीय आकाश
  • Coordinate axes — निर्देशांक अक्ष
  • Coordinate plane — निर्देशांक तल
  • Origin — मूल बिंदु
  • Octant — अष्टांश
  • Ordered triple — क्रमित त्रिक
  • Distance formula — दूरी सूत्र
  • Collinear — संरेख
  • Centroid — केंद्रक
  • Locus — बिंदुपथ
  • Perpendicular — लंब
  • Right-angled triangle — समकोण त्रिभुज

    • Common Mistakes

  • Confusing the meaning of coordinates: x is the distance from the YZ-plane, not from the x-axis.
  • Writing a point in the XZ-plane with a non-zero y-coordinate (it must be 0).
  • Mismatching octant sign patterns (e.g. calling (4, −2, 3) octant VIII instead of IV).
  • Forgetting to square negative differences correctly: (−3 − 2)² = 25, not −25.
  • Declaring collinearity from equal distances instead of checking PQ + QR = PR.

    • 💡 Key Learning Points

    • ✓Three mutually perpendicular axes (x, y, z) meeting at the origin define a rectangular Cartesian system in space.
    • ✓The XY-, YZ- and ZX-planes divide space into eight octants, each with its own sign pattern (I: +,+,+ to VIII: +,−,−).
    • ✓The coordinates (x, y, z) of a point are its distances from the YZ-, ZX- and XY-planes respectively.
    • ✓Points on axes and planes have zero coordinates in a fixed pattern, e.g. (x, 0, 0) on the x-axis and (x, y, 0) in the XY-plane.
    • ✓PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²] is used for collinearity, triangle classification and loci.

    👨‍🏫 Teaching Tips

    • →Use the corner of the classroom: two walls and the floor are the three coordinate planes and their edges are the axes.
    • →Hold up a box or a matchbox to locate a corner point by moving along x, then y, then z.
    • →Give students the octant sign list and play a quick 'name the octant' round with random points.
    • →Derive the 3D distance formula by applying Pythagoras twice (floor diagonal, then height).
    • →Show that the 2D distance formula is the special case z₁ = z₂.
    • →Assign an application: distance of a drone from its launch point given three coordinates.

    📋 Assessment Questions

    1. What are the y- and z-coordinates of a point on the x-axis?
    2. Name the octants of (4, −2, 3), (4, −2, −5) and (−2, −4, −7).
    3. Find the distance between (1, −3, 4) and (−4, 1, 2).
    4. Show that (−2, 3, 5), (1, 2, 3) and (7, 0, −1) are collinear.
    5. Find the equation of the set of points equidistant from (1, 2, 3) and (3, 2, −1).

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    Keep learning

    All chapters in Mathematics - Class 11

    1. 1. Sets
    2. 2. Relations and Functions
    3. 3. Trigonometric Functions
    4. 4. Complex Numbers and Quadratic Equations
    5. 5. Linear Inequalities
    6. 6. Permutations and Combinations
    7. 7. Binomial Theorem
    8. 8. Sequences and Series
    9. 9. Straight Lines
    10. 10. Conic Sections
    11. 11. Introduction to Three Dimensional Geometry
    12. 12. Limits and Derivatives
    13. 13. Statistics
    14. 14. Probability