Y-Axis 3D Rotation Matrix Calculator

Visualize and calculate 3D rotation around the Y-axis with ease. Enter the angle and see the rotation matrix and 3D representation instantly.

Rotation Matrix (Y-axis)

$$ R_y(\theta) = \begin{pmatrix} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \end{pmatrix} $$

3D Visualization

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Understanding Y-Axis Rotation Matrices

In 3D space, a rotation matrix around the Y-axis is used to transform vectors by rotating them about the Y-axis. This transformation is fundamental in computer graphics, robotics, and physics. The matrix is defined by a single angle, θ, and it alters the X and Z coordinates of a vector while keeping the Y coordinate unchanged.

The formula for a rotation matrix \(R_y\) around the Y-axis by an angle \( \theta \) is:

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

This calculator helps you generate this matrix for any given angle in degrees or radians and visualize its effect in 3D. Use it to understand spatial transformations and matrix operations in an intuitive way.