pöörlemismatriks

Pöörlemismatriks, also known as a rotation matrix, is a fundamental concept in linear algebra and geometry, used to represent rotations in Euclidean space. It is a square matrix that, when multiplied by a vector, rotates the vector by a specified angle in a specified plane. 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-dimensional space, a rotation matrix is defined as:

R(θ) = [cos(θ) -sin(θ)]

[sin(θ) cos(θ)]

where θ is the angle of rotation. This matrix rotates a vector counterclockwise by θ radians.

In three-dimensional space, rotation matrices can be more complex, as they can represent rotations about any axis. However, they still maintain the properties of being orthogonal and having a determinant of 1. Rotation matrices are widely used in computer graphics, robotics, and physics to model and simulate rotations in three-dimensional space.

The concept of rotation matrices can be extended to higher-dimensional spaces, although their practical applications become less intuitive. In general, a rotation matrix in n-dimensional space is an n x n orthogonal matrix with a determinant of 1.