Definition

1. Adjective:
   • Having a set of mutually perpendicular axes; meeting at right angles: In geometry, "orthogonal" describes lines, planes, or vectors that intersect at a 90-degree angle (a right angle). It is a fundamental concept in coordinate systems and spatial relationships.
   • Statistically unrelated: In mathematics and statistics, "orthogonal" describes variables or factors that are independent and uncorrelated, meaning there is no linear relationship between them.
   • Not pertinent to the matter under consideration; irrelevant or extraneous: In a more general or figurative sense, "orthogonal" can describe ideas, arguments, or facts that are completely separate from and unrelated to the main topic or issue.

Usage Examples

• Geometric Meaning:
  • The x-axis and y-axis in a standard graph are orthogonal to each other.
  • The architect designed the building with orthogonal walls, creating a grid-like floor plan.
• Statistical/Mathematical Meaning:
  • In the experiment, the two variables were found to be orthogonal, showing no correlation.
  • The researcher used orthogonal factors to simplify the complex data analysis.
• Figurative Meaning (Irrelevant):
  • His comment about the weather was completely orthogonal to our discussion on economic policy.
  • The lawyer objected, stating the witness's personal opinion was orthogonal to the facts of the case.

Advanced Usage

• "Orthogonal complement": In linear algebra, the set of all vectors orthogonal to every vector in a given subspace.
  • To solve the system, we first found the orthogonal complement of the given vector space.
• "Orthogonal transformation": A linear transformation that preserves the lengths of vectors and the angles between them, such as a rotation or reflection.
  • Applying an orthogonal transformation to the data set preserved the distances between all points.
• "Mutually orthogonal": Describing multiple items (e.g., vectors, dimensions) that are all perpendicular to each other.
  • In a three-dimensional Cartesian system, the x, y, and z axes are mutually orthogonal.

Variants and Related Words

• Orthogonality (noun): The state or property of being orthogonal.
  • The orthogonality of the two research approaches allowed them to be combined without conflict.
• Orthogonally (adverb): In an orthogonal manner.
  • The streets in the old town are arranged orthogonally, forming a perfect grid.

Synonyms

• Perpendicular: Meeting at a right angle (for the geometric sense).
• Uncorrelated: Having no mutual relationship or connection (for the statistical sense).
• Irrelevant: Not connected with or relevant to something (for the figurative sense).
• Extraneous: Irrelevant or unrelated to the subject being dealt with.

Related Phrases and Concepts

• Orthogonal projection: A projection of a vector onto a subspace such that the error vector (the difference) is orthogonal to the subspace.
  • We used an orthogonal projection to find the point on the plane closest to the given point.
• Orthogonal functions: A set of functions whose inner product (integral of their product) is zero, indicating a form of independence.
  • Fourier analysis relies on representing signals as sums of orthogonal functions like sines and cosines.
• Orthogonal matrix: In linear algebra, a square matrix whose inverse is equal to its transpose, representing an orthogonal transformation.
  • An orthogonal matrix preserves the dot product when it multiplies a vector.

Adjective

1. having a set of mutually perpendicular axes; meeting at right angles
   • wind and sea may displace the ship's center of gravity along three orthogonal axes
   • a rectangular Cartesian coordinate system
2. statistically unrelated
3. not pertinent to the matter under consideration
   • an issue extraneous to the debate
   • the price was immaterial
   • mentioned several impertinent facts before finally coming to the point