Orthogonal

TL;DR

• Describes a perpendicular (right-angle) relationship between two entities.
• Implies independence: changes in one do not affect the other.
• Appears in geometry, vector spaces, physics, engineering, and computer science.

Definition

Orthogonal is a term used to describe a relationship between two things that are perpendicular to each other. In this context, orthogonal means “at right angles to,” and is often used to describe a relationship between two things that are unrelated or independent of each other.

Explanation

• In geometry, orthogonal means two lines meet at a 90 degree angle.
• In multi-dimensional mathematics, two vectors are orthogonal when the angle between them is 90 degrees.
• Orthogonality is commonly used to express independence: when two things are orthogonal, changes in one do not affect the other.
• The concept is used across fields to separate complex systems into smaller, independent parts for analysis and design.

Examples

Orthogonal lines in geometry

In geometry, two lines are said to be orthogonal if they are perpendicular to each other. This means that the angle between the two lines is 90 degrees. For example, if we have two lines on a coordinate plane, one running horizontally (the x-axis) and one running vertically (the y-axis), these lines are orthogonal to each other. The x-axis and y-axis are used to create a coordinate system, which allows us to plot points on a two-dimensional plane. In this case, the x-axis and y-axis are independent of each other; changes in the x-coordinate do not affect the y-coordinate, and vice versa.

Orthogonal vectors in mathematics

In mathematics, two vectors are said to be orthogonal if they are perpendicular to each other in a multi-dimensional space. For example, consider two vectors in three-dimensional space, denoted by A and B. If the angle between these two vectors is 90 degrees, then the vectors are orthogonal to each other. In this case, the vectors are independent of each other; changes in one vector do not affect the other vector.

Complex system decomposition (car example)

One key aspect of orthogonality is that it allows us to analyze and understand complex systems by breaking them down into smaller, independent parts. For example, consider a car. A car is a complex system that is made up of many different parts, including the engine, transmission, wheels, suspension, and more. Each of these parts has its own function and interacts with the other parts in different ways. By analyzing the car in terms of its orthogonal parts, we can understand how the car works and how to fix it when it breaks down.

Use cases

• Mathematics: used to define and analyze functions and transformations such as the Fourier transform and the Laplace transform.
• Physics: describes relationships between different physical quantities, for example position and momentum.
• Engineering: describes relationships between different components of a system, such as the forces acting on a structure.
• Computer science: describes relationships between parts of a program or system, for example the relationship between the input and output of a function.

Notes or pitfalls

• Orthogonality is a tool for simplifying analysis by identifying independent components; treating non-orthogonal interactions as orthogonal would be misleading.
• The term is used both geometrically (perpendicularity) and conceptually (independence), but both usages share the core idea of “no influence” between the two entities.

Related terms

• Perpendicular
• Independent
• Vector
• Multi-dimensional space
• x-axis, y-axis
• Coordinate system
• Fourier transform
• Laplace transform
• Position and momentum
• Input and output (of a function)