Pöörlemismatrix

Pöörlemismatrix, also known as a rotation matrix, is a fundamental concept in linear algebra and geometry used to perform rotations in space. It is a special orthogonal matrix that preserves the length of vectors and the angles between them, making it ideal for rotating objects without distortion.

In two-dimensional space, a standard rotation matrix is expressed as:

[ cos θ -sin θ ]

[ sin θ cos θ ]

where θ represents the angle of rotation. When this matrix multiplies a vector, it rotates the vector counterclockwise by θ degrees or radians around the origin. For three-dimensional rotations, the rotation matrix becomes more complex, typically involving rotations around the x, y, or z axes, and can be represented using Euler angles or rotation axis-angle formulas.

Rotation matrices are widely used in computer graphics, robotics, aerospace, and physics to simulate and control rotational movements. They are also essential in algorithms for computer vision, animation, and navigation where precise manipulation of orientation is required.

Mathematically, properties of a rotation matrix include orthogonality (its transpose equals its inverse) and a determinant of +1. These properties ensure that the rotation preserves vector magnitudes and the overall structure of the space.

Overall, the pöörlemismatrix is a key tool for representing and calculating rotations in coordinate systems, enabling accurate modeling of rotational dynamics across various scientific and engineering disciplines.