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Chapter 11: Introduction to Three Dimensional Geometry

Class 11 • Mathematics • Chapter 11

Introduction to Three Dimensional Geometry

Adding a third axis so we can describe points anywhere in space, not just on a flat page.

CBSE / NCERT Aligned   School Revise

Chapter Roadmap

From 2D to 3D • Coordinate Planes • Octants • Coordinates of a Point • Signs in Each Octant • Distance Between Points • Key Results • Applications
1

Why Three Dimensional Geometry Matters

Everything you have plotted so far has lived on a flat plane, using just two numbers, x and y. But the real world has depth as well as width and height, so to locate a point in a room, in the sky, or in a 3D model on a computer, we need a third number. This chapter adds one more axis, the z-axis, and shows how three numbers (x, y, z) pin down any point in space.

Once we can name points in space, we can measure the straight-line distance between any two of them, just as we did on the plane, but now with three differences instead of two. The ideas here are the gateway to vectors, to the geometry of planes and lines in space that you meet in Class 12, and to fields like engineering, architecture and computer graphics, where everything is built in three dimensions.

Key idea
A point in space needs three coordinates (x, y, z). The three axes create three coordinate planes and divide all of space into eight regions called octants. The distance between two points uses all three differences under one square root.
2

Key Terms You Must Know
Term Meaning Example
Axes in space Three mutually perpendicular lines: the x-axis, y-axis and z-axis. They meet at the origin O
Coordinate planes The xy-plane, yz-plane and zx-plane formed by pairs of axes. The floor of a room is like the xy-plane
Octant One of the eight regions into which the three planes divide space. The first octant has all coordinates positive
Coordinates of a point The ordered triple (x, y, z) that locates a point in space. (2, 3, 5)
Origin The point where the three axes meet, with coordinates (0, 0, 0). O = (0, 0, 0)
Distance formula The straight-line distance between two points in space. √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
Projection Dropping a point onto a plane by setting one coordinate to 0. (2,3,5) projects to (2,3,0) on the xy-plane
3

Core Concepts, Step by Step

1. From Two Dimensions to Three

On the plane, two perpendicular axes, x and y, let us locate any point with two numbers. In space we add a third axis, the z-axis, perpendicular to both of the others, so the three axes meet at the origin like the corner of a room where two walls meet the floor. A point now needs three numbers (x, y, z): how far along, how far across, and how far up. We use a right-handed system, the standard convention in all of mathematics and physics.

2. The Three Coordinate Planes

Each pair of axes determines a flat plane. The x-axis and y-axis form the xy-plane, the y-axis and z-axis form the yz-plane, and the z-axis and x-axis form the zx-plane. A point lies in the xy-plane exactly when its z-coordinate is 0, in the yz-plane when its x-coordinate is 0, and in the zx-plane when its y-coordinate is 0. These three planes are the walls and floor against which every point is measured.

3. Coordinates of a Point in Space

To find the coordinates of a point P in space, measure its perpendicular distances to the three coordinate planes, taking signs into account. The result is an ordered triple (x, y, z). Here x is the distance from the yz-plane, y the distance from the zx-plane and z the distance from the xy-plane. The origin is (0, 0, 0); a point on the x-axis has the form (x, 0, 0), and a point in the xy-plane has the form (x, y, 0).

3D axes: a point P(x, y, z) and its distance from the origin

Three dimensional axes with point P and projection lines

4. The Eight Octants and Their Signs

The three coordinate planes slice space into eight regions called octants, just as the two axes split the plane into four quadrants. The first octant has all three coordinates positive. The other seven are found by changing the signs of one, two or all three coordinates. The table below lists the sign pattern of each octant so you can name the octant of any point at sight.

Signs of the coordinates in each octant

Octant x y z
I + + +
II − + +
III − − +
IV + − +
V + + −
VI − + −
VII − − −
VIII + − −

5. Distance Between Two Points

The straight-line distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]. It is the same idea as the distance formula on the plane, but with a third squared difference for the z-direction. The formula comes from applying Pythagoras twice, which we prove in the next section.

6. Distance From the Origin and From the Axes

Putting the origin (0, 0, 0) into the distance formula gives the distance of a point (x, y, z) from the origin as √(x² + y² + z²). With distances alone we can also test geometric facts: three points are collinear when the longest of the three pairwise distances equals the sum of the other two, and a triangle is right-angled when its three squared distances satisfy the Pythagoras relation.
4

Key Results & Proofs

The central result of this chapter is the distance formula. Everything else, including the distance from the origin and tests for collinearity, follows from it.

Result 1: The Distance Formula in Space

Statement. The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].

Proof

Apply Pythagoras twice: once on the base of a box, then to the diagonal.

Build a box (cuboid) with PQ as its long diagonal and edges along the axes.  –  set up a rectangular box
The edges have lengths (x₂−x₁), (y₂−y₁) and (z₂−z₁).  –  the three differences
The floor diagonal has length √[(x₂−x₁)² + (y₂−y₁)²] by Pythagoras.  –  Pythagoras on the base
d² = [(x₂−x₁)² + (y₂−y₁)²] + (z₂−z₁)² Pythagoras again, using the height
d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] take the square root

This is exactly the plane distance formula with one extra term for the z-direction.
Result 2: Distance From the Origin

Statement. The distance of the point (x, y, z) from the origin is √(x² + y² + z²).

Proof

This is the distance formula with one point taken as the origin.

Use the distance formula with the origin (0, 0, 0) as one point.  –  let P = origin
d = √[(x − 0)² + (y − 0)² + (z − 0)²] substitute
d = √(x² + y² + z²) simplify

So the point (3, 4, 12) is at distance √(9 + 16 + 144) = √169 = 13 from the origin.
Result 3: A Test for Collinearity

Statement. Three points are collinear if the largest of their three pairwise distances equals the sum of the other two.

Proof

On a straight line, the two shorter gaps add up to the whole.

Take three points A, B and C and find the distances AB, BC and CA.  –  compute all three distances
If the points lie on one straight line, one point lies between the other two.  –  picture them on a line
(longest distance) = (sum of the other two) the two shorter pieces make up the longest
If this holds, the three points are collinear; if not, they form a triangle.  –  the conclusion

For example AB + BC = AC means B lies on the segment from A to C.
5

Worked Examples
Example 1

Question: Name the octant in which the point (1, 2, 3) lies.

▶ Show full working

Check the sign of each coordinate against the octant table.

All three coordinates are positive: x > 0, y > 0, z > 0.  –  read off the signs
The all-positive pattern is the first octant.  –  match the table

Answer: Octant I.
Example 2

Question: Name the octant of the point (−1, 2, −3).

▶ Show full working

Find the sign pattern and match the table.

Signs are x negative, y positive, z negative, that is (−, +, −).  –  read off the signs
This pattern is octant VI.  –  match the table

Answer: Octant VI.
Example 3

Question: Write the coordinates of a point that lies on the x-axis.

▶ Show full working

A point on the x-axis has no y or z displacement.

On the x-axis both y and z are 0.  –  property of the x-axis
So the point has the form (x, 0, 0).  –  general form

Answer: Any point of the form (x, 0, 0), for example (5, 0, 0).
Example 4

Question: Find the distance of the point (2, 3, 6) from the origin.

▶ Show full working

Use d = √(x² + y² + z²).

d = √(2² + 3² + 6²) substitute
d = √(4 + 9 + 36) square each coordinate
d = √49 = 7 simplify

Answer: The distance is 7 units.
Example 5

Question: Find the distance between (1, −3, 4) and (4, 1, 4).

▶ Show full working

Use the distance formula in space.

d = √[(4−1)² + (1−(−3))² + (4−4)²] substitute the points
d = √[3² + 4² + 0²] work out the differences
d = √25 = 5 simplify

Answer: The distance is 5 units.
Example 6

Question: Find the distance between (−1, 3, −4) and (1, −3, 4).

▶ Show full working

Apply the distance formula carefully with the signs.

d = √[(1−(−1))² + (−3−3)² + (4−(−4))²] substitute
d = √[2² + (−6)² + 8²] work out the differences
d = √(4 + 36 + 64) = √104 add
d = 2√26 simplify the surd

Answer: The distance is 2√26 units.
Example 7

Question: Show that the points A(0, 7, 10), B(−1, 6, 6) and C(−4, 9, 6) form a right-angled isosceles triangle.

▶ Show full working

Find the three squared distances and check for equal sides and the Pythagoras relation.

AB² = (−1)² + (−1)² + (−4)² = 18 square distance A to B
BC² = (−3)² + 3² + 0² = 18 square distance B to C
AC² = (−4)² + 2² + (−4)² = 36 square distance A to C
AB² = BC², so two sides are equal (isosceles).  –  equal sides
AB² + BC² = 36 = AC², so the angle at B is a right angle.  –  Pythagoras holds

Answer: It is a right-angled isosceles triangle, right-angled at B.
Example 8

Question: Find the coordinates of the point where (2, 3, 4) lands when projected onto the xy-plane.

▶ Show full working

Projection onto the xy-plane sets the z-coordinate to 0.

On the xy-plane the z-coordinate is 0.  –  property of the xy-plane
So (2, 3, 4) projects to (2, 3, 0).  –  drop the z value

Answer: (2, 3, 0).
Example 9

Question: Find the distance of the point (3, 4, 12) from the origin.

▶ Show full working

Use d = √(x² + y² + z²).

d = √(3² + 4² + 12²) substitute
d = √(9 + 16 + 144) square each coordinate
d = √169 = 13 simplify

Answer: The distance is 13 units.
Example 10

Question: Are the points A(1, 2, 3), B(2, 3, 4) and C(3, 4, 5) collinear?

▶ Show full working

Find the three distances and check whether the longest equals the sum of the other two.

AB = √(1 + 1 + 1) = √3 distance A to B
BC = √(1 + 1 + 1) = √3 distance B to C
AC = √(4 + 4 + 4) = 2√3 distance A to C
AB + BC = √3 + √3 = 2√3 = AC.  –  the longest equals the sum

Answer: Yes, the three points are collinear.
Example 11

Question: Name the octant of the point (5, −2, −7).

▶ Show full working

Find the sign pattern and match the table.

Signs are (+, −, −).  –  read off the signs
This pattern is octant VIII.  –  match the table

Answer: Octant VIII.
Example 12

Question: A point P(x, y, z) is equidistant from A(1, 2, 3) and B(3, 2, 1). Find the relation between its coordinates.

▶ Show full working

Set PA² = PB² and simplify.

PA² = (x−1)² + (y−2)² + (z−3)² square distance to A
PB² = (x−3)² + (y−2)² + (z−1)² square distance to B
−2x − 6z + 10 = −6x − 2z + 10 set PA² = PB² and cancel
4x = 4z collect terms
x = z the relation

Answer: The relation is x = z.
6

Where You Meet This in Real Life

Maps and GPS

Locating a point on Earth, or an aircraft in the sky, needs three numbers: two for position and one for height or depth.

Architecture and engineering

Buildings, bridges and machine parts are designed in three dimensions, with every joint given coordinates in space.

Computer graphics and games

Every object in a 3D game or film is built from points with (x, y, z) coordinates that the computer moves and projects onto the screen.

Robotics

A robot arm reaches a point in space by setting the three coordinates of its gripper, then computing distances to move.

Medical imaging

Scans such as MRI build a three-dimensional picture of the body from points measured throughout space.
7

Practice Sets A to D

Practice Set A – Basics

A1. Name the octant of the point (2, 3, 4).

▶ Reveal full working

Check the signs.

All coordinates are positive.
That is octant I.

Answer: Octant I.

A2. Write the coordinates of the point on the z-axis at height 5.

▶ Reveal full working

On the z-axis both x and y are 0.

A point on the z-axis has the form (0, 0, z).
At height 5 it is (0, 0, 5).

Answer: (0, 0, 5).

A3. Find the distance between the origin and (1, 2, 2).

▶ Reveal full working

Use √(x² + y² + z²).

d = √(1 + 4 + 4) substitute
d = √9 = 3 simplify

Answer: 3 units.

A4. Which coordinate is zero for a point in the yz-plane?

▶ Reveal full working

Recall which axis is missing from that plane.

The yz-plane is built from the y-axis and z-axis.
A point on it has x = 0.

Answer: The x-coordinate is 0.

Practice Set B – Conceptual

B1. Into how many regions do the three coordinate planes divide space?

▶ Reveal full working

Compare with the four quadrants of the plane.

Two axes give four quadrants in the plane.
Three planes give eight regions in space.

Answer: Eight octants.

B2. What is the condition for a point to lie in the xy-plane?

▶ Reveal full working

Think about the missing direction.

The xy-plane has no height above or below it.
So a point on it has z = 0.

Answer: Its z-coordinate is 0.

B3. Name the three coordinate planes.

▶ Reveal full working

Each is formed by a pair of axes.

They are the xy-plane, the yz-plane and the zx-plane.

Answer: The xy-plane, yz-plane and zx-plane.

B4. For a point on the x-axis, which coordinates are zero?

▶ Reveal full working

A point on the x-axis has no spread in the other directions.

Both the y-coordinate and the z-coordinate are 0.

Answer: y and z are both 0.

Practice Set C – Application / Numerical

C1. Find the distance between (1, 2, 3) and (4, 5, 6).

▶ Reveal full working

Use the distance formula.

d = √[(3)² + (3)² + (3)²] work out the differences
d = √27 = 3√3 simplify

Answer: 3√3 units.

C2. Find the distance between (2, −1, 3) and (2, 3, 3).

▶ Reveal full working

Only the y-coordinate changes.

d = √[0² + 4² + 0²] work out the differences
d = √16 = 4 simplify

Answer: 4 units.

C3. Name the octant of the point (−3, −4, 5).

▶ Reveal full working

Find the sign pattern.

Signs are (−, −, +).
This is octant III.

Answer: Octant III.

C4. Show that the point (3, 4, 12) is at distance 13 from the origin.

▶ Reveal full working

Use the distance-from-origin formula.

d = √(9 + 16 + 144) substitute
d = √169 = 13 simplify

Answer: The distance is 13, as required.

Practice Set D – HOTS / Word Problems

D1. Show that the points A(−2, 3, 5), B(1, 2, 3) and C(7, 0, −1) are collinear.

▶ Reveal full working

Find the three distances and check that the longest equals the sum of the other two.

AB = √(9 + 1 + 4) = √14 distance A to B
BC = √(36 + 4 + 16) = 2√14 distance B to C
AC = √(81 + 9 + 36) = 3√14 distance A to C
AB + BC = √14 + 2√14 = 3√14 = AC.  –  the longest equals the sum

Answer: They are collinear.

D2. Find the point on the x-axis that is equidistant from A(1, 2, 3) and B(3, 2, 1).

▶ Reveal full working

A point on the x-axis is (x, 0, 0); set PA² = PB².

(x−1)² + 4 + 9 = (x−3)² + 4 + 1 set the squared distances equal
−2x + 14 = −6x + 14 expand and cancel
4x = 0 collect terms
x = 0 solve

Answer: The point is the origin (0, 0, 0).

D3. A point P(x, y, z) is equidistant from A(2, 0, 0) and B(0, 2, 0). Find the relation between its coordinates.

▶ Reveal full working

Set PA² = PB² and simplify.

(x−2)² + y² + z² = x² + (y−2)² + z² set squared distances equal
−4x + 4 = −4y + 4 expand and cancel
x = y the relation

Answer: The relation is x = y.

D4. Find the distance of the point (a, b, c) from the x-axis.

▶ Reveal full working

The nearest point on the x-axis is (a, 0, 0).

The foot of the perpendicular on the x-axis is (a, 0, 0).  –  drop to the axis
d = √[(a−a)² + b² + c²] distance to that foot
d = √(b² + c²) simplify

Answer: √(b² + c²).

Chapter Summary

Everything in One Glance

Three Axes

Space needs three perpendicular axes (x, y, z) meeting at the origin (0, 0, 0).

  

Coordinate Planes

The xy-, yz- and zx-planes; a point is on a plane when the missing coordinate is 0.
 

Octants

The three planes divide space into eight octants, the first having all coordinates positive.

  

Coordinates

A point is fixed by an ordered triple (x, y, z); axes and planes have one or more zero coordinates.
 

Distance

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

  

From the Origin

Distance of (x, y, z) from the origin is √(x² + y² + z²).
 
8

Are You Exam-Ready?

8-Point Exam Quick-Check

1 Name the octant of the point (4, −1, 2).
 
2 Write the coordinates of a point in the zx-plane.
 
3 Find the distance of (6, 8, 0) from the origin.
 
4 Find the distance between (1, 0, 0) and (0, 1, 0).
 
5 Which coordinate is zero for a point on the y-axis besides y?
 
6 How many octants are there in three dimensional space?
 
7 Project the point (3, −2, 5) onto the xy-plane.
 
8 State the distance formula between two points in space.
 

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Class 11 Mathematics Chapter 11: Introduction to Three Dimensional Geometry, Complete Notes and Practice

These free Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry notes follow the latest NCERT 2026-27 syllabus and give complete, exam-ready coverage for board exams and competitive foundations. Includes worked examples, step-by-step proofs and graded practice, free on SchoolRevise.com.
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