Class 11 Maths Notes

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Three Dimensional Geometry Class 11 Notes

Complete, exam-ready notes on introduction to three-dimensional geometry: how the three coordinate axes divide space into octants, coordinates of a point in 3D, the distance and section formulas, direction cosines and direction ratios of a line, and the angle between two lines using direction ratios — written for CBSE boards and JEE revision.

Class:11Subject:MathematicsCovers:CBSE · JEE

7 Key Formulas4 Practice MCQs

DWritten byDeep Narayan

UpdatedSeptember 2026

Key Concept Summary

What is this chapter about in one line?

Three-dimensional geometry extends the coordinate plane into space: you locate points with three coordinates and measure distances, ratios and angles using direction cosines.

Coordinate Axes and Planes in Space

Definition

Coordinate axes

Three mutually perpendicular lines — the x-axis, y-axis and z-axis — meet at the origin O(0, 0, 0). Any point P in space is located by the ordered triple (x, y, z).

The xy-plane, yz-plane and zx-plane divide space into eight octants.
In the first octant, x > 0, y > 0 and z > 0.
The octant sign pattern (±, ±, ±) identifies all eight octants — a frequent MCQ topic.
The z-coordinate extends the 2D plane vertically into 3D space.

Distance Between Two Points

Definition

3D distance formula

The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is: $P Q = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$ . This is the natural extension of the 2D distance formula with the third dimension added.

Definition

Distance from origin

For any point P(x, y, z), the distance from the origin O(0, 0, 0) is: $O P = x^{2} + y^{2} + z^{2}$ .

Section Formula

Definition

Internal division

If point R divides the segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally in the ratio m : n, the coordinates of R are: $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n}, \frac{m z _{2} + n z _{1}}{m + n})$ .

Definition

Midpoint

The midpoint (m = n) of segment PQ is: $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})$ . This is a special case of the section formula with m = n = 1.

External division: replace m + n with m − n in the denominator, and flip the sign of n.
The section formula in 3D mirrors the 2D version — just add the z-component.

Direction Cosines and Direction Ratios

Definition

Direction cosines

The cosines of the angles α, β, γ that a line makes with the positive x-, y- and z-axes are called direction cosines, denoted l, m, n. They always satisfy: $l^{2} + m^{2} + n^{2} = 1$ .

Definition

Direction ratios

a, b, c

Any three numbers a, b, c proportional to the direction cosines are direction ratios of the line: $l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}, m = \frac{b}{a ^{2} + b ^{2} + c ^{2}}, n = \frac{c}{a ^{2} + b ^{2} + c ^{2}}$ . Direction ratios are not unique — any scalar multiple works.

Practical use

For a line through P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), the direction ratios are simply (x₂ − x₁, y₂ − y₁, z₂ − z₁). Normalise to get direction cosines.

Angle Between Two Lines

Definition

Angle using direction cosines

If two lines have direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂), the angle θ between them satisfies: $cos θ = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$ . For the acute angle, take the absolute value.

Definition

Angle using direction ratios

For direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂): $cos θ = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$ .

Lines are parallel when their direction ratios are proportional: a₁/a₂ = b₁/b₂ = c₁/c₂.
Lines are perpendicular when a₁a₂ + b₁b₂ + c₁c₂ = 0.

Collinearity of Three Points

Definition

Collinearity test

Three points A(x₁, y₁, z₁), B(x₂, y₂, z₂) and C(x₃, y₃, z₃) are collinear if and only if their direction ratios are proportional — equivalently, $\frac{x _{2} - x _{1}}{x _{3} - x _{1}} = \frac{y _{2} - y _{1}}{y _{3} - y _{1}} = \frac{z _{2} - z _{1}}{z _{3} - z _{1}}$ . If all three ratios are equal, the points lie on a single line.

An equivalent check: AB + BC = AC (sum of distances).
If the ratios are not equal, the three points form a triangle with non-zero area in 3D.
Collinearity is a prerequisite for applying the section formula — ensure the points are collinear before dividing a segment.

Solved Examples

Example: Find the distance between the points P(1, −3, 4) and Q(3, −5, 2).

Solution: PQ = √[(3 − 1)² + (−5 + 3)² + (2 − 4)²] = √[4 + 4 + 4] = √12 = 2√3 units.

Example: Find the direction cosines of the line joining A(1, 2, −3) and B(3, −2, 1).

Solution: Direction ratios = (3−1, −2−2, 1−(−3)) = (2, −4, 4). The magnitude is √(4 + 16 + 16) = √36 = 6. So l = 2/6 = 1/3, m = −4/6 = −2/3, n = 4/6 = 2/3. Check: l² + m² + n² = 1/9 + 4/9 + 4/9 = 1.

NCERT Exercise Questions

From the textbook

All NCERT questions for this chapter — NCERT Maths Part I, Chapter 11 (Introduction to Three Dimensional Geometry). Questions follow the textbook order so you can match them to the book.

Exercises (NCERT Q1–Q9)

1) A point is on the x-axis. What are its y-coordinate and z-coordinates?
2) A point is in the XZ-plane. What can you say about its y-coordinate?
3) Name the octants in which the following points lie: (1, 2, 3), (4, −2, 3), (4, −2, −5), (4, 2, −5), (−4, 2, −5), (−4, 2, 5), (−3, −1, 6), (−2, −4, −7).
4) Fill in the blanks: (i) The x-axis and y-axis taken together determine a plane known as _______. (ii) The coordinates of points in the XY-plane are of the form _______. (iii) Coordinate planes divide the space into _______ octants.
5) Find the distance between the following pairs of points: (i) (2, 3, 5) and (4, 3, 1), (ii) (−3, 7, 2) and (2, 4, −1), (iii) (−1, 3, −4) and (1, −3, 4), (iv) (2, −1, 3) and (−2, 1, 3).
6) Show that the points (−2, 3, 5), (1, 2, 3) and (7, 0, −1) are collinear.
7) Verify the following: (i) (0, 7, −10), (1, 6, −6) and (4, 9, −6) are the vertices of an isosceles triangle. (ii) (0, 7, 10), (−1, 6, 6) and (−4, 9, 6) are the vertices of a right angled triangle. (iii) (−1, 2, 1), (1, −2, 5), (4, −7, 8) and (2, −3, 4) are the vertices of a parallelogram.
8) Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, −1).
9) Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (−4, 0, 0) is equal to 10.

Answers at a glance

1
Q1 → y-coordinate = 0 and z-coordinate = 0.
2
Q2 → The y-coordinate is 0.
3
Q3 → I, IV, VIII, V, VI, II, III, VII respectively.
4
Q4 → (i) XY-plane, (ii) (x, y, 0), (iii) eight (8).
5
Q5 → (i) 2√5, (ii) √43, (iii) 2√26, (iv) 2√5.
6
Q6 → Collinear — shown by AB + BC = AC.
7
Q7 → (i) isosceles, (ii) right angled, (iii) parallelogram — all verified.
8
Q8 → x − 2z = 0 (the perpendicular bisector plane).
9
Q9 → 9x² + 25y² + 25z² = 225.

CBSE Pattern Questions

How CBSE asks this chapter

CBSE Class 11 Mathematics board paper: Section A (20 MCQs × 1M), Section B (5 × 2M), Section C (6 × 3M) and Section D (4 × 4M). Introduction to Three Dimensional Geometry gives 1-mark MCQs on coordinates on the axes and the distance formula, 2/3-mark questions with midpoints, section formula and distances, and 4-mark questions proving geometric facts about triangle/centroid in 3D.

MCQ Questions

A point on the y-axis in 3D has coordinates

The distance between (1, 2, 3) and (2, 3, 4) is

The distance of the point (1, 2, 3) from the origin is

The midpoint of the segment joining (1, 2, 3) and (3, 4, 5) is

1 Mark Questions

1
Write the coordinates of the origin. (Ans: (0, 0, 0))
2
State the distance formula between (x₁, y₁, z₁) and (x₂, y₂, z₂).
3
What are the coordinates of a point on the z-axis?
4
Write the section formula for the point dividing (x₁,y₁,z₁) and (x₂,y₂,z₂) in the ratio m : n.

2 Mark Questions

1
Find the distance between the points (2, 3, 5) and (4, 3, 1). (Ans: √20 = 2√5)
2
Find the midpoint of the segment joining (−1, 3, 4) and (3, 1, 6). (Ans: (1, 2, 5))
3
Find the distance of the point (3, −4, 5) from the origin. (Ans: 5√2)

3 Mark Questions

1
Show that the points (0, 1, 2), (2, 1, 0) and (1, 1, 1) form an isosceles triangle by computing the three distances. (Ans: two distances equal √8)
2
Find the coordinates of the centroid of the triangle with vertices (0, 0, 0), (0, 3, 0) and (0, 0, 6) and justify it lies on the z-plane they form.
3
Find the point on the x-axis which is equidistant from (2, 1, 0) and (0, 1, 2). (Ans: (1, 0, 0) — solve (x−2)² + 1 = x² + 5)

4 Mark Questions

1
Prove the distance formula in 3D by extending the 2D result, then use it to show the triangle with vertices (1, 2, 3), (4, 6, 3) and (6, 7, 3) is right-angled by the converse of Pythagoras.
2
Derive the section formula for internal division in the ratio m : n for two points in space, and use it to find the coordinates of the point that divides (1, 1, 1) and (7, 4, 7) in the ratio 2 : 1. (Ans: (5, 3, 5))

PDF: Assignments & Tests

Printable study material

Free downloadable PDFs for Introduction to Three Dimensional Geometry — two topic-wise assignments covering every concept of the chapter, and two full CBSE-pattern tests (each 90 minutes, 40 marks) mirroring the board paper. No signup required.

Free PDF Downloads & Assignments

No Signup Required

Practice path: complete Assignment 1 first (topic-by-topic), then Assignment 2 (mixed difficulty with full solutions), then attempt the timed tests.

3D Geometry Assignment 1 — Topic-wise Questions

Covers all six topics: coordinate axes and planes in space; distance between two points; section formula; direction cosines and direction ratios; angle between two lines; and collinearity of three points.

introduction-to-three-dimensional-geometry-class-11-notes-assignment-1.pdf

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3D Geometry Assignment 2 — Challenge + Integrated Problems

Higher-difficulty set with multi-step problems (parallelogram proofs, equidistant points, external division, direction cosines, angles between lines, collinearity) and full worked answers. Includes an answer key.

introduction-to-three-dimensional-geometry-class-11-notes-assignment-2.pdf

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3D Geometry Test 1 — CBSE Pattern (90 min, 40 marks)

Full board-pattern test: Section A (MCQ/VSA 1M × 10), Section B (2M × 4), Section C (3M × 4), Section D (5M × 2) = 40 marks. Time: 90 minutes. Includes answer key.

introduction-to-three-dimensional-geometry-class-11-notes-test-1-cbse.pdf

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3D Geometry Test 2 — CBSE Pattern (90 min, 40 marks)

Second full paper with different questions, same 40-mark / 90-minute blueprint. Includes answer key.

introduction-to-three-dimensional-geometry-class-11-notes-test-2-cbse.pdf

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Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Distance between two points

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}

Internal section formula

(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n}, \frac{m z _{2} + n z _{1}}{m + n})

Midpoint formula

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})

Direction cosine identity

l^{2} + m^{2} + n^{2} = 1

Direction ratios to cosines

l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}

Angle between two lines

cos θ = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}

Distance from origin

x^{2} + y^{2} + z^{2}

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

Direction cosines satisfy l² + m² + n² = 1 always — use this to verify your answer.
Direction ratios are not unique; only their ratios matter.
The 3D section formula mirrors 2D — just add the z-component.
Parallel lines: direction ratios are proportional. Perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0.
Octant identification from sign of coordinates is a common MCQ topic.
Distance from origin is a special case: √(x² + y² + z²).
For angle between lines, normalise direction ratios to cosines first or use the full formula.
The midpoint formula is the section formula with m = n = 1 — know both.

FAQ

Frequently asked questions

What are direction cosines?

Direction cosines are the cosines of the angles a line makes with the positive x-, y- and z-axes, denoted l, m, n. They always satisfy l² + m² + n² = 1.

How do you find the distance between two points in 3D?

Use the extended distance formula: PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²], which adds the z-component to the 2D version.

What is the section formula in 3D?

A point dividing the segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) in ratio m : n has coordinates ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)).

How do direction ratios differ from direction cosines?

Direction ratios are any three numbers proportional to the direction cosines — they are not unique. Direction cosines are the normalised values that satisfy l² + m² + n² = 1.

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