Coordinate Axes and Coordinate Planes in 3D space

Last Updated : 20 Jun, 2026

In three-dimensional space (ℝ³), we also consider height or depth. Therefore, a point is represented as (x, y, z), where x, y and z determine the exact position of the point in space. These values are called the coordinates of the point.

In 3D coordinate geometry, we use three coordinate axes to locate a point in space.

Coordinate Axes

Three mutually perpendicular axes intersect at a point called the origin (O). These axes are

1


  • X-axis: measures left-right direction
  • Y-axis: measures front-back direction
  • Z-axis: measures up-down direction

A point in 3D space is written as P(x, y, z)

For example, P(2, 3, 4) means:

  • Move 2 units along the X-axis,
  • 3 units along the Y-axis,
  • 4 units along the Z-axis.

Coordinate Planes

In three-dimensional space, the planes formed by the coordinate axes are called coordinate planes: the XY-plane, YZ-plane, and ZX-plane. Each plane is formed by two axes and is perpendicular to the third axis. These three planes divide the space into eight regions called octants.

2-

In this system:

  • The origin is written as (0, 0, 0).
  • Any point on the x-axis is of the form (x, 0, 0).
  • Any point on the y-axis is of the form (0, y, 0).
  • Any point on the z-axis is of the form (0, 0, z).
  • Any point on the XY-plane is of the form (x, y, 0).
  • Any point on the YZ-plane is of the form (0, y, z).
  • Any point on the ZX-plane is of the form (x, 0, z).

Distance between Two Points 

Consider two points (x1, y1, z1) and (x2, y2, z2) in three-dimensional space. The distance between them is found using an extension of the distance formula from two dimensions.

The distance between the two points is given by:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

Sample Problems

Question 1: Let's say we have a point on the x-axis, what is its y-coordinate and z-coordinate? 

Solution: In the figure, the point lies on x-axis. It can be noticed that it's coordinates for y and z are equal to zero. 

Question 2: 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. 

Question 3: Calculate the distance between (0,0,0) and (5,4,3). 

Solution: 

For the points (x1, y1, z1) and (x2, y2, z2) 

\sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 - y_3)^2}

Here, (x1, y1, z1) = (0, 0, 0) and (x2, y2, z2) = (5, 4, 3). Let the distance be "l" 

l = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 - y_3)^2}

= \sqrt{(5 - 0)^2 + (4 - 0)^2 + (3 - 0)^2}

= \sqrt{25 + 16 + 9}

= \sqrt{50}

= 5√2 

Question 4: Calculate the distance between (0,0,0) and (1,2,3). 

Solution: 

For the points (x1, y1, z1) and (x2, y2, z2) 

\sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 - y_3)^2}

Here, (x1, y1, z1) = (0,0,0) and (x2, y2, z2) = (1,2,3). Let the distance be "l" 

l = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 - y_3)^2}

= \sqrt{1 + 4 + 9}

= \sqrt{1 + 9 + 4}

= \sqrt{14}

Question 5: Calculate the distance between (1,1,1) and (2,4,3). 

Solution: 

For the points  (x1, y1, z1) and (x2, y2, z2) 

\sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 - y_3)^2}

Here, (x1, y1, z1) = (1,1,1) and (x2, y2, z2) = (2,4,3). Let the distance be "l" 

l = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 - y_3)^2}

= \sqrt{(2 - 1)^2 + (4 - 1)^2 + (3 - 1)^2}

= \sqrt{1^2 + 3^2 + 2^2}

= \sqrt{14}    

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