Geometric Bites

2×2 Orthogonal Matrix Mastery — A Generalised Construction Orthogonal matrices in two dimensions reveal one of the cleanest structures in linear algebra. A 2×2 matrix is orthogonal when its columns (and rows) satisfy two conditions: They are perpendicular (their dot product is zero); They have unit length (their magnitude is one). This article presents a clear generalisation: any pair of perpendicular vectors with equal magnitude can be normalised to form an orthogonal matrix. 1. Begin with two perpendicular vectors Let the first vector be: (a, b) A perpendicular vector can be chosen as: (−b, a) Their dot products confirm orthogonality: (a, b) · (−b, a) = −ab + ab = 0 (a, −b) · (b, a) = ab − ab = 0 2. Compute their shared magnitude Both vectors have the same length: |(a, b)| = √(a² + b²) We can therefore normalise each one by dividing by √(a² + b²). 3. Form the matrix using the normalised vectors Place the two normalised vectors...

Orthogonal Matrices and Mutually Orthogonal Vectors Orthogonal matrices appear naturally throughout linear algebra, geometry, physics, and computer graphics. They preserve lengths, angles, and orientation, which makes them fundamental in describing rotations and rigid motions in three-dimensional space. This article provides a clear and carefully structured explanation of what orthogonal matrices are, why they matter, and how to verify that a given matrix is orthogonal. 1. Definition of an Orthogonal Matrix Let M be an n × n square matrix. M is called orthogonal if it satisfies: M M T = I Here: M T is the transpose of M. I is the identity matrix of the same size. Because of this property, every orthogonal matrix has a very useful consequence: M -1 = M T This means that the inverse of an orthogonal matrix is obtained simply by transposing it. This property is central to rigid-body transformations in 3D geometry and computer graphics. 2...