Rotatsioonimaatriks

A rotation matrix is a fundamental concept in linear algebra and geometry, used to represent rotations in a coordinate system. 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 particularly useful in computer graphics, robotics, and physics for transforming coordinates and objects in space.

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

R(θ) = [cos(θ) -sin(θ)]

[sin(θ) cos(θ)]

In three dimensions, rotations can occur around any axis. The rotation matrix for a rotation by an angle θ around the z-axis (the axis perpendicular to the xy-plane) is:

Rz(θ) = [cos(θ) -sin(θ) 0]

[sin(θ) cos(θ) 0]

[ 0 0 1]

For rotations around the x-axis and y-axis, similar matrices can be derived. Rotation matrices can also be composed to represent more complex rotations by applying multiple rotations sequentially.

One important property of rotation matrices is that they are orthogonal, meaning their inverse is equal to their transpose. This property ensures that the length of vectors is preserved during rotation, which is a key characteristic of rotations in Euclidean space.

Rotation matrices are a powerful tool for manipulating and transforming geometric data, and their properties make them well-suited for a wide range of applications in mathematics and engineering.