Rotatsioonimatiise

Rotatsioonimatiise, also known as rotational matrices, are mathematical tools used primarily in linear algebra and geometry to describe rotations in Euclidean space. These matrices represent linear transformations that rotate vectors in a coordinate system around an origin. In two-dimensional space, a rotation matrix rotates a point (x, y) around the origin by a specified angle θ, transforming it into a new position (x′, y′). The matrix takes the form:

[ cos(θ) -sin(θ) ]

[ sin(θ) cos(θ) ]

When applied to a vector, the new coordinates are calculated by multiplying the matrix with the original vector. Rotational matrices are orthogonal, meaning their inverse is equal to their transpose, which simplifies computations involving rotations.

In three-dimensional space, rotations become more complex due to the additional axis of rotation. A general rotation matrix in 3D can be constructed using Euler angles, quaternions, or axis-angle representation. For example, a rotation around the z-axis by angle θ is represented similarly to the 2D case, while rotations around the x or y axes introduce additional terms to account for the third dimension. These matrices are essential in computer graphics, robotics, and physics simulations, where objects must be manipulated or oriented in space.

Rotational matrices preserve lengths and angles, ensuring that geometric properties remain consistent after transformation. They are also used in solving problems involving rigid body dynamics, where the orientation of objects must be tracked over time. The properties of rotational matrices make them fundamental in both theoretical and applied mathematics, enabling precise and efficient modeling of rotational motion.