Introduction to 3D Geometry – Class 11 Maths | Coordinates, Distance Formula, Examples

Introduction

Geometry is one of the oldest branches of mathematics, and the study of two-dimensional (2D) figures such as lines, circles, and conic sections has already been covered in earlier classes. However, the real world around us is three-dimensional. Every object we see – a book, a building, or even our own bodies – has length, breadth, and height. To mathematically describe positions and shapes in such a space, we need Three Dimensional (3D) Geometry.

This chapter introduces the very first concepts of 3D coordinate geometry, where every point is represented by three numbers instead of two. The chapter covers:

The concept of coordinate axes and planes in 3D.
How to assign coordinates to a point in space.
The formula to calculate the distance between two points in 3D.

Though the chapter appears short, it is extremely important because it lays the groundwork for advanced 3D geometry studied in Class 12, including direction ratios, equations of lines and planes, shortest distance between skew lines, and vector approach to 3D geometry.

In competitive exams like JEE Main, KCET, and COMEDK, questions from this section are usually formula-based, making it a scoring topic. If mastered properly, students can solve such questions in less than a minute during the exam.

11.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space

The Need for a Third Axis

In 2D coordinate geometry, we represent a point with an ordered pair (x, y). The x-coordinate gives its distance from the y-axis, while the y-coordinate gives its distance from the x-axis. However, this only tells us the position of the point on a flat surface (the xy-plane).

To extend this to real-world 3D space, we introduce a third axis, the z-axis, perpendicular to both the x-axis and y-axis. Now, every point can be uniquely described by three numbers.

Definition of the Three Axes

x-axis: A horizontal line in space (extends left and right).
y-axis: Another horizontal line, but perpendicular to the x-axis (extends forward and backward).
z-axis: A vertical line perpendicular to both x and y axes (extends up and down).

All three intersect at the origin (0, 0, 0). Together, they form a system of mutually perpendicular lines called the Rectangular Cartesian Coordinate System in 3D.

The Origin

The point where x, y, and z axes intersect is called the origin (O). It acts as the reference point for locating every other point in space.

The Coordinate Planes

Just as two perpendicular axes in 2D divide the plane into four quadrants, in 3D the three axes define three planes:

XY-plane (z = 0): The plane formed by x- and y-axes.
YZ-plane (x = 0): The plane formed by y- and z-axes.
ZX-plane (y = 0): The plane formed by z- and x-axes.

These three planes intersect at the origin and divide the entire space into eight octants (similar to quadrants in 2D, but doubled due to the third dimension).

Octants

The first octant is where x, y, and z are all positive.
Moving across planes changes the signs of the coordinates.
Thus, each point (x, y, z) lies in one of the eight octants depending on the signs of x, y, and z.

11.3 Coordinates of a Point in Space

Ordered Triplets

Any point in space can be represented as P(x, y, z), called an ordered triplet.

x: The perpendicular distance of the point from the YZ-plane.
y: The perpendicular distance of the point from the ZX-plane.
z: The perpendicular distance of the point from the XY-plane.

Thus, the coordinates give us the exact position of the point in 3D.

Sign Convention

If the point lies on the positive side of an axis, the coordinate is positive.
If the point lies on the negative side, the coordinate is negative.

For example:

(2, –3, 5) lies in the octant where x > 0, y < 0, and z > 0.

Special Points

Origin: (0, 0, 0)
x-axis points: (x, 0, 0)
y-axis points: (0, y, 0)
z-axis points: (0, 0, z)
xy-plane: (x, y, 0)
yz-plane: (0, y, z)
zx-plane: (x, 0, z)

11.4 Distance between Two Points

The 2D Distance Formula

In two dimensions, the distance between A(x₁, y₁) and B(x₂, y₂) is given by:

AB = √((x₂ – x₁)² + (y₂ – y₁)²)

This comes from the Pythagoras theorem.

The 3D Distance Formula

In three dimensions, for points A(x₁, y₁, z₁) and B(x₂, y₂, z₂):

AB = √((x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²)

This too is derived using the Pythagoras theorem, extended to 3D space.

Example 1

Find the distance between A(1, 2, 3) and B(4, 6, 8).

Solution:
AB = √((4 – 1)² + (6 – 2)² + (8 – 3)²)
= √(9 + 16 + 25) = √50 = 5√2

Solved Examples

Example 2

Find the coordinates of a point which lies in the first octant and is 4 units from YZ-plane, 5 units from ZX-plane, and 6 units from XY-plane.

Solution:

Distance from YZ-plane = x = 4
Distance from ZX-plane = y = 5
Distance from XY-plane = z = 6
Thus, the coordinates are (4, 5, 6).

Example 3

Find the distance between P(–2, 3, 1) and Q(1, –1, –4).

Solution:
PQ = √((1 – (–2))² + (–1 – 3)² + (–4 – 1)²)
= √((3)² + (–4)² + (–5)²)
= √(9 + 16 + 25)
= √50 = 5√2

Applications of 3D Coordinate Geometry

Physics:
- Representing the motion of particles in space.
- Describing positions and velocities using vectors.
Engineering & Architecture:
- Designing 3D structures, bridges, and buildings.
- CAD (Computer-Aided Design).
Computer Graphics:
- 3D modeling in animations and video games.
Astronomy:
- Locating celestial bodies using spatial coordinates.
Exams:
- Direct formula-based questions in JEE, KCET, COMEDK.

Common Pitfalls

Forgetting the correct sign convention (x, y, or z negative).
Mixing up coordinate planes (e.g., thinking xy-plane means x = 0).
Applying the 2D distance formula in 3D problems.
Ignoring octants when interpreting coordinates.

FAQs

Q1. How many octants are there in 3D geometry?
There are eight octants, depending on the sign of x, y, and z coordinates.

Q2. What are the coordinates of the origin?
Origin is at (0, 0, 0).

Q3. What is the difference between quadrants and octants?
Quadrants are in 2D geometry (four regions), while octants are in 3D (eight regions).

Q4. What is the formula for distance between two points in 3D?
√((x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²)

Q5. Is this topic important for JEE Main?
Yes. Usually, 1–2 questions are asked (straightforward formula-based).

Marks Allotment in Exams

CBSE Boards: 4–6 marks (direct application of distance formula).
JEE Main: 1–2 questions (8 marks).
KCET/COMEDK: 1–2 direct MCQs (scoring).

Conclusion

The Introduction to Three Dimensional Geometry is the foundation stone for all of 3D coordinate geometry. This chapter introduces the three axes, planes, coordinates of a point in space, and the formula for the distance between two points.

Mastering this topic ensures:

Ability to visualize in 3D.
Accuracy in distance calculations.
Readiness for advanced 3D geometry in Class 12.

For competitive exams, this chapter is considered scoring and straightforward. Since questions are formula-based, students should practice enough problems to gain speed and avoid sign mistakes.