Matrix Z-Axis Rotation

Calculator for the rotation of a 3x3 matrix around the Z axis

Z-Axis Rotation Calculator

Instructions

Enter the rotation angle for rotation around the Z-axis.

  • Unit: Switch between degrees or radians
  • Mode: Active (rotate object) or Passive (rotate coordinates)
Z-Axis Rotation
Rotation Matrix Rz(α)
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Z-Axis Rotation - Overview

Z-Axis Rotation

The Z-axis rotation (Yaw) rotates objects or coordinates around the Z-axis. The matrix rotation distinguishes between active and passive rotation.

Rotation Matrix Formula

The general formula for rotation around the Z-axis:

\(R_z(\alpha) = \begin{bmatrix} \cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0 & 0 & 1 \end{bmatrix}\)

Example: 90° Rotation

Active rotation (counterclockwise):

\(R_z(90°) = \begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix}\)

Rotation Modes
  • Active Rotation: Rotates the vector/object in the coordinate system (counterclockwise). Also called geometric transformation.
  • Passive Rotation: Rotates the coordinate system (clockwise). The vector remains unchanged.


Description of Matrix Z-Axis Rotation

Active Rotation

With active rotation, the vector or the object is rotated in the coordinate system. The active rotation is also called a geometric transformation. The rotation is counterclockwise.

Example: 90° Rotation of the Z-axis

Active Z-rotation formula 1

Active Z-rotation formula 2

Properties
  • Axis: Rotation around Z-axis (Yaw)
  • Direction: Counterclockwise (when looking along positive Z-axis)
  • Third row: Always [0, 0, 1] (Z-component unchanged)
  • Orthogonal: RT = R-1
  • Determinant: det(R) = 1
  • 2D-like: Essentially a 2D rotation in the XY-plane

Passive Rotation

With passive rotation, the coordinate system is rotated. The vector remains unchanged. The rotation is clockwise.

Example: 90° Rotation of the Z-axis

Passive Z-rotation formula 1

Passive Z-rotation formula 2

Matrix Structure

Active Rotation (counterclockwise):

\(R_z(\alpha) = \begin{bmatrix} \cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0 & 0 & 1 \end{bmatrix}\)

Passive Rotation (clockwise):

\(R_z^{-1}(\alpha) = \begin{bmatrix} \cos \alpha & \sin \alpha & 0\\ -\sin \alpha & \cos \alpha & 0\\ 0 & 0 & 1 \end{bmatrix}\)

Practical Applications

Aerospace & Robotics:

  • Aircraft yaw control (left/right)
  • Drone heading control
  • Robotic base rotation
  • Satellite yaw axis control

Computer Graphics & Gaming:

  • Camera pan (looking left/right)
  • Character turning animations
  • Top-down game rotations
  • Compass orientation
Key Characteristics

The Z-axis rotation matrix is unique in that its third row and third column remain mostly constant [0, 0, 1], meaning the Z-component of any vector is unchanged during rotation. Only the X and Y components are affected. This rotation is commonly called "Yaw" in aerospace applications and represents left/right turning. It's essentially a 2D rotation extended to 3D space.

Related Rotations

For complete 3D orientation, you may need to combine rotations around multiple axes:

  • X-axis rotation (Roll): Barrel roll rotation
  • Y-axis rotation (Pitch): Nose up/down rotation
  • Combined rotations: Use Euler angles or quaternions for full 3D rotation