Rotationsmatriser

Rotationsmatriser, also known as rotation matrices, are fundamental tools in linear algebra and geometry, used to represent rotations in Euclidean space. They are square matrices that, when multiplied by a vector, rotate the vector by a specified angle around a specified axis. In two-dimensional space, a rotation matrix is a 2x2 matrix, while in three-dimensional space, it is a 3x3 matrix.

The general form of a 2D rotation matrix is:

[ 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 around any axis. The general form of a 3D rotation matrix around the z-axis is:

[ cos(θ) -sin(θ) 0 ]

[ sin(θ) cos(θ) 0 ]

[ 0 0 1 ]

Rotation matrices have several important properties. They are orthogonal, meaning their transpose is equal to their inverse, and they preserve the dot product and the length of vectors. They are also used in computer graphics, robotics, and other fields where rotations are a key component.