pöörlemismaatriksiks

Pöörlemismaatriksiks, or rotation matrix in English, is a mathematical tool used in linear algebra and geometry to represent the rotation of a coordinate system around a fixed point. It is a square matrix that encodes the direction cosines between two coordinate systems. In a two-dimensional space, a rotation matrix for an angle θ is represented as:

[cos(θ) -sin(θ)]

[sin(θ) cos(θ)]

In a three-dimensional space, the rotation matrix for an angle θ around an axis defined by a unit vector (u_x, u_y, u_z) is given by:

[cos(θ) + u_x^2(1 - cos(θ)) u_x*u_y*(1 - cos(θ)) - u_z*sin(θ) u_x*u_z*(1 - cos(θ)) + u_y*sin(θ)]

[u_y*u_x*(1 - cos(θ)) + u_z*sin(θ) cos(θ) + u_y^2(1 - cos(θ)) u_y*u_z*(1 - cos(θ)) - u_x*sin(θ)]

[u_z*u_x*(1 - cos(θ)) - u_y*sin(θ) u_z*u_y*(1 - cos(θ)) + u_x*sin(θ) cos(θ) + u_z^2(1 - cos(θ))]

Rotation matrices are orthogonal matrices, meaning their inverse is equal to their transpose. They preserve the length of vectors and the angles between them, making them useful in computer graphics, physics simulations, and other fields that involve spatial transformations. The product of two rotation matrices corresponds to the composition of the two rotations they represent.