Geometric Bites

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...

The Difference Between the Lines 𝐀 + t𝐁 and 𝐁 + t(𝐀 − 𝐁) A line in vector form is defined by two components: a base point that determines its position, and a direction vector that determines its orientation. Two expressions may involve the same vectors but still represent completely different lines when either the base point or the direction vector changes. The expressions L₁: 𝐀 + t𝐁 L₂: 𝐁 + t(𝐀 − 𝐁) provide a clear example of how distinct lines arise from different vector components. 1. Line L₁: 𝐀 + t𝐁 The expression 𝐀 + t𝐁 describes a line passing through the point represented by vector 𝐀 with direction vector 𝐁. As the real parameter t varies, the expression generates all points on the line. Base point: 𝐀 Direction vector: 𝐁 This is the line through 𝐀 directed along 𝐁. 2. Line L₂: 𝐁 + t(𝐀 − 𝐁) The expression 𝐁 + t(𝐀 − 𝐁) describes a different line. Its base point is 𝐁, and its direction vector is t...