Rotationsmatricen

Rotationsmatricen, also known as rotation matrices, are mathematical tools used to describe the orientation of an object in space or to perform a rotation in Euclidean space. In two dimensions, a rotation matrix represents a rotation around the origin by a certain angle. The general form of a 2D rotation matrix for a counterclockwise rotation by an angle theta is given by: [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]]. When this matrix is multiplied by a vector representing a point in the 2D plane, the resulting vector represents the rotated point.

In three dimensions, rotation matrices are 3x3 matrices. They can represent rotations around the x, y, or z axes, or rotations around an arbitrary axis. A rotation around the z-axis by an angle theta has the form: [[cos(theta), -sin(theta), 0], [sin(theta), cos(theta), 0], [0, 0, 1]]. Similar matrices exist for rotations around the x and y axes. Combining rotations in 3D typically involves multiplying their respective rotation matrices.

Rotation matrices have several important properties. They are orthogonal matrices, meaning their transpose is equal to their inverse. This ensures that rotations preserve lengths and angles. The determinant of a rotation matrix is always 1, indicating that it preserves volume and orientation. These matrices are fundamental in fields such as computer graphics, robotics, physics, and engineering, where they are used to model and manipulate objects undergoing rotational transformations.