vectorrotation

Vector rotation is a fundamental operation in geometry and linear algebra that involves changing the orientation of a vector without altering its magnitude. Imagine a vector as an arrow originating from a point; rotating it means spinning that arrow around a fixed point, typically the origin, by a specified angle.

In two-dimensional space, a vector (x, y) can be rotated counterclockwise by an angle θ around the origin using a rotation matrix. The new coordinates (x', y') are given by x' = x cos(θ) - y sin(θ) and y' = x sin(θ) + y cos(θ). This transformation can be expressed concisely as a matrix multiplication:

[ x' ] = [ cos(θ) -sin(θ) ] [ x ]

[ y' ] [ sin(θ) cos(θ) ] [ y ]

In three-dimensional space, rotation is more complex as it requires specifying an axis of rotation in addition to the angle. Common methods include using Euler angles or quaternions, which provide different ways to represent and perform rotations. For instance, rotating around one of the coordinate axes (x, y, or z) can be achieved using specific 3x3 rotation matrices.

Vector rotation is crucial in various fields, including computer graphics for orienting objects, robotics for controlling joint movements, physics for describing motion, and image processing for image manipulation. Understanding vector rotation is essential for anyone working with spatial transformations and geometric computations.