Rotasjonsmatrisen

Rotasjonsmatrisen, also known as the 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.

The 2D rotation matrix is a 2x2 matrix that rotates vectors in the xy-plane. It is defined as:

[ cos(θ) -sin(θ) ]

[ sin(θ) cos(θ) ]

where θ is the angle of rotation. This matrix can be derived using the properties of trigonometry and the dot product.

In 3D space, rotations can occur around any axis. The 3D rotation matrix around the z-axis is similar to the 2D case, but extended to 3 dimensions. The general form of a 3D rotation matrix around an arbitrary axis can be derived using the Rodrigues' rotation formula or Euler's rotation theorem.

Rotasjonsmatrisen is widely used in computer graphics, robotics, and physics for simulating rotations and transformations. It is also used in signal processing for rotating coordinate systems and in machine learning for data transformation.

The determinant of a rotation matrix is always 1, indicating that rotations preserve volume. Additionally, the transpose of a rotation matrix is equal to its inverse, meaning that rotating a vector and then rotating it back results in the original vector.