Matrix Rotation Simulator
Visualize how a 2D rotation matrix moves vectors around the origin
Rotation Matrix Formula
R(θ) = [[cos θ, -sin θ], [sin θ, cos θ]]
Multiplying a vector by this matrix rotates the vector by θ around the origin. The determinant is 1, so area and orientation are preserved.
⚠Angles are interpreted in degrees for input, then converted to radians for cosine and sine calculations.
How to Read the Simulator
Gray Vector
Original vector before rotation.
Red Vector
Rotated vector after matrix multiplication.
Basis Arrows
Blue and green arrows show rotated basis directions.
Area Scale
A rotation has determinant 1, so area is unchanged.
Example: A 90 degree rotation maps (x, y) to (-y, x). So (3, 1) becomes (-1, 3).
Applications
Linear AlgebraComputer GraphicsRoboticsCoordinate Transforms
Frequently Asked Questions
What is a 2D rotation matrix?▼
A 2D rotation matrix is [[cos theta, -sin theta], [sin theta, cos theta]]. It rotates vectors around the origin by angle theta.
Does a rotation matrix change vector length?▼
No. A pure rotation preserves vector length, angles, area, and orientation. Its determinant is 1.
What does a positive rotation angle mean?▼
Using the standard coordinate convention, a positive angle rotates counterclockwise around the origin.
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