Pööremaatriks

Pööremaatriks, also known as a rotation matrix, is a fundamental concept in linear algebra and is used extensively in computer graphics, physics, and engineering. It is a matrix that performs a rotation on a vector or coordinate system. In two dimensions, a rotation matrix can be represented as a 2x2 matrix. When multiplied by a vector, it transforms the vector to a new position representing a rotation around the origin by a specified angle.

The general form of a 2D rotation matrix that rotates points in the counterclockwise direction by an angle theta is given by:

[[cos(theta), -sin(theta)],

[sin(theta), cos(theta)]]

In three dimensions, rotation matrices are 3x3 matrices and are more complex as rotations can occur around any of the three principal axes (x, y, or z), or indeed any arbitrary axis. Rotations in 3D are not commutative, meaning the order in which rotations are applied matters. Common 3D rotation matrices rotate around the x, y, or z axes by a specific angle.

Rotation matrices possess specific properties. They are orthogonal matrices, meaning their transpose is equal to their inverse. This property ensures that rotations preserve the length and orientation of vectors. The determinant of a rotation matrix is always 1, signifying that rotations preserve volume. These matrices are crucial for transforming coordinates between different orientations, simulating movement in virtual environments, and analyzing physical systems undergoing rotational motion.