Matrix Y-Axis Rotation

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

Y-Axis Rotation Calculator

Instructions

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

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

Y-Axis Rotation

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

Rotation Matrix Formula

The general formula for rotation around the Y-axis:

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

Example: 90° Rotation

Active rotation (counterclockwise):

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

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 Y-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 Y-axis

Active Y-rotation formula 1

Active Y-rotation formula 2

Properties
  • Axis: Rotation around Y-axis (Pitch)
  • Direction: Counterclockwise (when looking along positive Y-axis)
  • Second row: Always [0, 1, 0] (Y-component unchanged)
  • Orthogonal: RT = R-1
  • Determinant: det(R) = 1

Passive Rotation

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

Example: 90° Rotation of the Y-axis

Passive Y-rotation formula 1

Passive Y-rotation formula 2

Matrix Structure

Active Rotation (counterclockwise):

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

Passive Rotation (clockwise):

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

Practical Applications

Aerospace & Robotics:

  • Aircraft pitch control (nose up/down)
  • Drone elevation control
  • Robotic arm elbow rotation
  • Satellite pitch axis control

Computer Graphics & Gaming:

  • Camera pitch (looking up/down)
  • Object tilt animations
  • Character nod animations
  • Flight simulators pitch control
Key Characteristics

The Y-axis rotation matrix is unique in that its second row remains constant [0, 1, 0], meaning the Y-component of any vector is unchanged during rotation. Only the X and Z components are affected. This rotation is commonly called "Pitch" in aerospace applications and represents nose-up or nose-down rotation. Note the asymmetry in the formula compared to X and Z rotations.

Related Rotations

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

  • X-axis rotation (Roll): Barrel roll rotation
  • Z-axis rotation (Yaw): Left/right rotation
  • Combined rotations: Use Euler angles or quaternions for full 3D rotation