What is a 2D rotation matrix?

A 2D rotation matrix is a 2×2 matrix [[cos θ, −sin θ], [sin θ, cos θ]] that rotates any point in the plane by angle θ around the origin, counterclockwise by convention.

Is the rotation clockwise or counterclockwise?

The standard rotation matrix rotates counterclockwise for positive angles. To rotate clockwise, enter a negative angle or transpose the matrix (swap sin and −sin positions).

Does rotating change the distance from the origin?

No. A rotation matrix is orthogonal (its columns are unit vectors and are perpendicular), so it preserves the length of every vector. The rotated point lies on the same circle of radius √(x²+y²).

How do I rotate around a point other than the origin?

Translate the point so the center of rotation becomes the origin, apply the rotation matrix, then translate back. Concretely: subtract the center, rotate, then add the center.

What is the inverse of a rotation matrix?

The inverse of R(θ) is R(−θ), which equals the transpose of R(θ). This means you can reverse a rotation simply by negating the angle or transposing the matrix.

Rotation Matrix 2D

Enter an angle and a point to get the 2D rotation matrix and the transformed coordinates.

Rotation Parameters

Angle (θ)

Input Point (x, y)

Rotation Matrix R(θ)

〈

—

〉

[ [cos θ, −sin θ], [sin θ, cos θ] ]

Rotated Point

—

Distance from origin: —

Step-by-step

Enter values and click Calculate.

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Summary

A 2D rotation matrix transforms a point by rotating it around the origin by a given angle. Given angle θ, the rotation matrix is [[cos θ, −sin θ], [sin θ, cos θ]]. Multiply this matrix by a column vector [x, y] to get the rotated point. This tool computes the matrix, applies it to your input point, and shows every step of the calculation.

How it works

Enter the rotation angle in degrees (or switch to radians).
Enter the x and y coordinates of the point to rotate.
The tool computes cos(θ) and sin(θ) to build the 2×2 rotation matrix.
It multiplies the rotation matrix by the input vector [x, y].
The rotated point and the full matrix are displayed instantly.

Use cases

Verify hand-computed rotation matrix entries for a linear algebra problem.
Rotate a vector or point as part of a computer graphics pipeline.
Check rotated coordinates when implementing 2D game physics.
Understand the relationship between angle, cosine, and sine in transformations.
Convert between standard and rotated coordinate frames in robotics.
Compose multiple rotations by chaining the resulting matrices.

Frequently Asked Questions

Last updated: 2026-06-18 · Reviewed by Nham Vu