Pöörlemismaatriksitel

Pöörlemismaatriksitel, also known as rotation matrices, are fundamental tools in linear algebra and geometry, used to represent rotations in Euclidean space. They are square matrices that, when multiplied by a vector, rotate the vector by a specified angle around a specified axis. Rotation matrices are orthogonal, meaning their transpose is equal to their inverse, and their determinant is 1, indicating that they preserve the volume of the space they operate on.

In two dimensions, a rotation matrix is a 2x2 matrix that rotates vectors counterclockwise by an angle θ around the origin. The general form of a 2D rotation matrix is:

[ cos(θ) -sin(θ) ]

[ sin(θ) cos(θ) ]

In three dimensions, rotation matrices can represent rotations around any axis. The general form of a 3D rotation matrix around the z-axis by an angle θ is:

[ cos(θ) -sin(θ) 0 ]

[ sin(θ) cos(θ) 0 ]

[ 0 0 1 ]

Rotation matrices are widely used in computer graphics, robotics, and physics for tasks such as transforming coordinates, simulating rigid body motion, and modeling physical systems. They are also used in signal processing for tasks such as filtering and modulation.