Coordinate System in Three-Dimensional Geometry | NCERT Solutions | Solved Examples, FAQs

Class 11 Mathematics | Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS

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What is Three-Dimensional Geometry?

Three-Dimensional Geometry (3D Geometry) is a branch of mathematics that deals with points, lines, planes, and shapes in a space that has three dimensions—length, breadth, and height (or depth). Unlike two-dimensional geometry, which only deals with flat surfaces (like squares or circles), 3D Geometry helps us analyze objects in the real world, such as cubes, spheres, cylinders, and cones.

In the context of Class 11 Mathematics, this topic forms the foundation for advanced concepts in calculus, physics, engineering, and computer graphics.

1. Coordinates in 3D Space

In three-dimensional space, any point is represented by an ordered triplet (x,y,z), where:

• x-coordinate represents the position along the X-axis (length)
• y-coordinate represents the position along the Y-axis (breadth)
• z-coordinate represents the position along the Z-axis (height or depth)

The origin O(0,0,0) is the point where all three axes intersect.

Coordinate System in Three-Dimensional Geometry (three-dimensional space)

Coordinate system in 3D geometry (three-dimensional space) is based on three mutually perpendicular axes (coordinate) — x-axis, y-axis, and z–axis. It is used to find the location of any point in space.

The below figure shows the coordinate axes and planes in three dimensions.

Coordinate Planes in Three Dimensional Space

The planes XOY, YOZ, and ZOX are called XY-plane, YZ-plane, and ZX-plane respectively. The intersection of all the planes is called the origin.  

These are also known as the three coordinate planes. These planes divide the 3-D space into 8 octants. 

These octants are labelled as XOYZ, X′OYZ, X′OY′Z, XOY′Z, XOYZ′, X′OYZ′, X′OY′Z′ and XOY′Z′ and are respectively denoted by I, II, III, IV, V< VI, VII, and VIII quadrants.

Coordinates of a Point in Space

In 3D geometry, the coordinates of a point P is written in the form of P(x, y, z), where x, y and z are the distances of the point, from the YZ, ZX and XY-planes.

• Coordinates of any point at the origin is (0,0,0)
• Coordinates of any point on the x-axis is in the form of (x,0,0)
• Coordinates of any point on the y-axis is in the form of (0,y,0)
• Coordinates of any point on the z-axis is in the form of (0,0,z)
• Coordinates of any point on the XY-plane is in the form (x, y, 0)
• Coordinates of any point on the YZ-plane is in the form (0, y, z)
• Coordinates of any point on the ZX-plane is in the form (x, 0, z)

Sign of Coordinates in Different Octants:

The sign (+ or -) of the coordinates of a point determines the octant in which the point lies and is tabulated below:

Octants	I	II	III	IV	V	VI	VII	VIII
Co-ordinates
x	+	–	–	+	+	–	–	+
y	+	+	–	–	+	+	–	–
z	+	+	+	+	–	–	–	–

NCERT Solutions

1. A point is on the x-axis. What are its y-coordinate and z-coordinates?

Solution:

If a point is on the x-axis, then the coordinates of y and z are 0.

So the point is (x, 0, 0).

2. A point is in the XZ-plane. What can you say about its y-coordinate?

Solution:

If a point is in the XZ plane, then its y-co-ordinate is 0.

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).

Solution:

Here is the table which represents the octants:

Octants	I	II	III	IV	V	VI	VII	VIII
x	+	–	–	+	+	–	–	+
y	+	+	–	–	+	+	–	–
z	+	+	+	+	–	–	–	–

(i) (1, 2, 3)

Here, x is positive, y is positive, and z is positive.

So, it lies in the I octant.

(ii) (4, -2, 3)

Here, x is positive, y is negative, and z is positive.

So, it lies in the IV octant.

(iii) (4, -2, -5)

Here, x is positive, y is negative, and z is negative.

So, it lies in the VIII octant.

(iv) (4, 2, -5)

Here, x is positive, y is positive, and z is negative.

So, it lies in the V octant.

(v) (-4, 2, -5)

Here, x is negative, y is positive, and z is negative.

So, it lies in VI octant.

(vi) (-4, 2, 5)

Here, x is negative, y is positive, and z is positive.

So, it lies in the II octant.

(vii) (-3, -1, 6)

Here, x is negative, y is negative, and z is positive.

So, it lies in the III octant.

(viii) (2, -4, -7)

Here, x is positive, y is negative, and z is negative.

So, it lies in the VIII octant.

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.

Solution:

(i) The x-axis and y-axis, taken together, determine a plane known as XY Plane.

(ii) The coordinates of points in the XY-plane are of the form (x, y, 0).

(iii) Coordinate planes divide the space into eight octants.

FAQs on Coordinate Axes and Coordinate Planes

What is the XY-plane in 3D space?

The XY-plane is the plane where the Z-coordinate is always zero. It extends infinitely in the directions of the X and Y axes.

What are the three coordinate axes in 3D space?

The three coordinate axes in 3D space are the X-axis (horizontal), Y-axis (vertical), and Z-axis (depth).

How do you determine if a point lies on a specific coordinate plane?

A point lies on a specific coordinate plane if the coordinate corresponding to the axis perpendicular to that plane is zero.

Fill in the blanks: 

1. X and Y axis together make _____ plane. 
2. All the coordinate planes divide the 3d space into _______ octants. 

Answer: 

1. X and Y axis together make XY plane. 

2. All the coordinates planes divide the 3-D space into eight octants. 

Applications of 3D Geometry

• Understanding 3D models in physics and engineering
• Computer graphics and animation
• Robotics and architecture
• Navigation and GPS systems

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Summary

Three-Dimensional Geometry is a crucial topic in Class 11 Mathematics that prepares students for higher studies in mathematics, engineering, physics, and related fields. It enhances spatial understanding and helps in solving real-life problems involving three-dimensional spaces.

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