The 3D rotation is different from 2D rotation. In 3D Rotation we also have to define the angle of Rotation with the axis of Rotation.

For Example-Let us assume,

The initial coordinates of an object = (x0, y0, z0)

The Initial angle from origin = β

The Rotation angle = θ

The new coordinates after Rotation = (x1, y1, z1

In Three-dimensional plane we can define Rotation by following three ways

1. X-axis Rotation: We can rotate the object along x-axis. We can rotate an object by using following equation-

X1 = x0

Y1 = y0 cosθz0 x sinθ

Z1 = y0 x sinθ+ z0 x cosθ

We can represent 3D rotation in the form of matrix-

2. Y-axis Rotation: We can rotate the object along y-axis. We can rotate an object by using following equation-

x1 = z0 x sinθ + x0 x cosθ

y1 = y0

z1 = y0 x cosθ x0 x sinθ

We can represent 3D rotation in the form of matrix

3. Z-axis Rotation: We can rotate the object along z-axis. We can rotate an object by using following equation-

x1 = x0 x cosθ y0 x sinθ

y1 = x0 x sinθ + y0 x cosθ

z1 = z0

We can represent 3D rotation in the form of matrix

Example: A Point has coordinates P (2, 3, 4) in x, y, z-direction. The Rotation angle is 90 degrees. Apply the rotation in x, y, z direction, and find out the new coordinates of the point?

Solution: The initial coordinates of point = P (x0, y0, z0) = (2, 3, 4)

Rotation angle (θ) = 90°

For x-axis

Let the new coordinates = (x1, y1, z1) then,

x1= x0 = 2

y1= y0 x cosθ – z0 x sinθ  = 3 x cos90°– 4 x sin90° = 3 x 0 – 4 x 1 = -4

z1= y0 x sinθ  + z0 x cosθ  = 3 x sin90°+ 4 x cos90° = 3 x 1 + 4 x 0 = 3

The new coordinates of point = (2, -4, 3)

For y-axis

Let the new coordinates = (x1, y1, z1) then,

X1= z0 x sinθ + x0 x cosθ  = 4 x sin90° + 2 x cos90° = 4 x 1 + 2 x 0 = 4

y1= y0 = 3

z1= y0 x cos&ampθ  – x0 x sinθ  = 3 x cos90°– 2 x sin90° = 3 x 0 – 4 x 0 = 0

The new coordinates of point = (4, 3, 0)

For z-axis

Let the new coordinates = (x1, y1, z1) then,

x1= x0 x cosθ  – y0 x sinθ = 2 x cos90° – 3 x sin90° = 2 x 0 + 3 x 1 = 3

y1= x0 x sinθ  + y0 x cosθ  = 2 x sin90° ­+ 3 x cos90° = 2 x 1 + 3 x 0 = 2

z1= z0 =4

The New Coordinates of points = (3, 2, 4)