Coordinates of a Point in 3D Space – Class 11 Maths | Definition, Examples & Applications

Introduction

In 2D coordinate geometry, we used an ordered pair (x, y) to represent the position of a point in the plane. Each coordinate tells us how far the point is from one axis. But when moving into three dimensions (3D), every point requires three numbers — its distances from the three coordinate planes.

Thus, a point in 3D space is represented by an ordered triplet (x, y, z). These three numbers uniquely determine the location of the point in relation to the x-, y-, and z-axes. This is a fundamental idea in 3D coordinate geometry.

Defining the Coordinates of a Point

Consider a point P in 3D space. To locate it, we do the following:

Drop a perpendicular from P onto the XY-plane. Let the foot of the perpendicular be M.
Drop a perpendicular from M onto the x-axis. Let it meet the x-axis at A.

Now:

OA = x (distance along x-axis)
AM = y (distance along y-axis)
PM = z (distance along z-axis)

Thus, the point P is represented as P(x, y, z).

Geometric Interpretation

x: Distance of the point from the YZ-plane.
y: Distance of the point from the ZX-plane.
z: Distance of the point from the XY-plane.

So, each coordinate gives the distance of the point from a coordinate plane rather than from the axes directly.

Sign Convention

Just as in 2D geometry, the signs of coordinates in 3D depend on whether the point lies in the positive or negative direction of the axis:

If the point is to the right of YZ-plane, x is positive; if to the left, x is negative.
If the point is to the front of ZX-plane, y is positive; if behind, y is negative.
If the point is above XY-plane, z is positive; if below, z is negative.

This leads to the classification of points into eight octants depending on the signs of x, y, and z.

Coordinates of Special Points

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

Examples of Coordinates in 3D

P(3, –2, 5): Lies 3 units right of YZ-plane, 2 units behind ZX-plane, 5 units above XY-plane.
Q(0, 4, –3): Lies in the YZ-plane (since x = 0), 4 units ahead along y, and 3 units below XY-plane.
R(–5, 0, 0): Lies on the negative x-axis.

Relation with Octants

The position of the point (x, y, z) depends on the signs of x, y, and z:

( + , + , + ) → First octant
( – , + , + ) → Second octant
( – , – , + ) → Third octant
( + , – , + ) → Fourth octant
( + , + , – ) → Fifth octant
( – , + , – ) → Sixth octant
( – , – , – ) → Seventh octant
( + , – , – ) → Eighth octant

Algebraic Properties of Coordinates

If a point lies on x-axis, y = 0 and z = 0.
If a point lies on y-axis, x = 0 and z = 0.
If a point lies on z-axis, x = 0 and y = 0.
If a point lies in xy-plane, z = 0.
If a point lies in yz-plane, x = 0.
If a point lies in zx-plane, y = 0.

These properties are frequently tested in competitive exams.

Distance from Coordinate Planes

For a point P(x, y, z):

Distance from YZ-plane = |x|
Distance from ZX-plane = |y|
Distance from XY-plane = |z|

Solved Examples

Example 1: Identifying a Point

Find the coordinates of a point that is 4 units from YZ-plane, 5 units from ZX-plane, and 7 units from XY-plane, lying in the first octant.

Solution:

Distance from YZ-plane = x = 4
Distance from ZX-plane = y = 5
Distance from XY-plane = z = 7
So, point = (4, 5, 7).

Example 2: On Coordinate Plane

A point has coordinates (0, –3, 2). On which coordinate plane does it lie?

Solution:
x = 0 → lies on YZ-plane.

Example 3: Reflection in Planes

Find the coordinates of the image of point (3, –2, 5) with respect to:
(a) XY-plane
(b) YZ-plane
(c) ZX-plane

Solution:

(a) Reflection in XY-plane → (3, –2, –5)
(b) Reflection in YZ-plane → (–3, –2, 5)
(c) Reflection in ZX-plane → (3, 2, 5)

Example 4: Distance from Coordinate Plane

Find the distance of point A(–4, 3, –5) from:
(a) YZ-plane
(b) ZX-plane
(c) XY-plane

Solution:

(a) |x| = 4
(b) |y| = 3
(c) |z| = 5

Example 5: Symmetry in Octants

If a point P(2, –3, 4) is in one octant, write the coordinates of points symmetric to P with respect to all three coordinate planes.

Solution:

Symmetric in XY-plane → (2, –3, –4)
Symmetric in YZ-plane → (–2, –3, 4)
Symmetric in ZX-plane → (2, 3, 4)

Applications

Physics: Representing position and motion of particles in space.
Computer Graphics: 3D modeling and simulations use coordinates (x, y, z).
Engineering: Structural design in 3D using coordinates.
Astronomy: Locating stars and satellites in space.
Robotics: Robot motion planning in three dimensions.

Common Mistakes

Confusing distance from an axis with distance from a plane.
Forgetting sign convention of coordinates in different octants.
Assuming coordinates (x, y, z) mean distances along axes instead of distances from planes.
Misidentifying coordinate planes (e.g., mixing xy-plane with yz-plane).

Exam Weightage

CBSE Boards: 3–4 marks (direct questions on locating points).
JEE Main: 1 question (~4 marks, usually with distance from planes or reflections).
KCET/COMEDK: 1 MCQ (straightforward sign or coordinate identification).

Practice Problems

Identify the octant of (–2, 5, –3).
A point lies 3 units from YZ-plane, 4 units from ZX-plane, and 2 units from XY-plane. Find its coordinates in the second octant.
Write coordinates of points symmetric to (1, –2, 3) with respect to:
(a) XY-plane, (b) YZ-plane, (c) ZX-plane.
Find the coordinates of the point lying on the xy-plane at a distance 5 from the origin.
A point is 7 units from the x-axis, 6 units from the y-axis, and lies on the xy-plane. Find its coordinates.

FAQs

Q1. What is meant by coordinates of a point in 3D?
It is an ordered triplet (x, y, z) giving distances of the point from the three coordinate planes.

Q2. What is the difference between (x, y) in 2D and (x, y, z) in 3D?
In 2D, only two numbers are needed; in 3D, three are required to fix the position.

Q3. Can a point lie on more than one coordinate plane?
Yes. For example, (0, 0, z) lies on both YZ-plane and ZX-plane.

Q4. How do you determine the octant of a point?
Check the signs of x, y, and z.

Q5. Which coordinate is zero if the point lies on xy-plane?
The z-coordinate.

Conclusion

The concept of coordinates of a point in space is fundamental to 3D geometry. By extending from ordered pairs to ordered triplets (x, y, z), we can now describe any point in three-dimensional space.

Understanding sign conventions, coordinate planes, and octants is crucial for solving problems quickly in exams. This topic forms the base for distance formulas, section formulas, equations of lines and planes, and vectors, all of which are essential in higher-level mathematics and competitive exams.