rotatsioonimatriks

A "rotatsioonimatriks" is a mathematical construct used in linear algebra to represent rotations in Euclidean space. It is a square matrix whose primary function is to perform a rotation transformation on vectors within the space, preserving their length and the angles between them. Rotatsioonimatriksid are commonly applied in fields such as computer graphics, robotics, physics, and engineering to model rotational motion.

In two-dimensional space, a rotation matrix is a 2x2 matrix of the form:

\[ R(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \]

where \(\theta\) is the rotation angle. When this matrix multiplies a 2D vector, it rotates the vector counterclockwise by \(\theta\) degrees around the origin.

In three-dimensional space, rotation matrices are 3x3 and can represent rotations around any of the coordinate axes (x, y, z). For example, a rotation around the z-axis by an angle \(\theta\) is given by:

\[ R_z(\theta) = \begin{bmatrix}

\cos \theta & -\sin \theta & 0 \\

\sin \theta & \cos \theta & 0 \\

0 & 0 & 1 \end{bmatrix} \]

Rotation matrices are orthogonal and have a determinant of 1, which ensures that they preserve distances and orientation. They satisfy the property \( R(\theta) R(\phi) = R(\theta + \phi) \), meaning successive rotations can be combined through matrix multiplication.

Overall, rotatsioonimatriksid provide a fundamental means of describing and computing rotations in vector spaces, essential for precise geometric transformations in various scientific and technological applications.