Orthogonality
Two vectors are orthogonal if their dot product is zero, which generalises the geometric notion of perpendicularity to any vector space equipped with an inner product. A set of vectors is orthogonal if every pair of distinct vectors in the set is orthogonal, and if additionally every vector has unit length, the set is called orthonormal. Orthogonality is fundamental in Fourier series (decomposing signals into orthogonal frequency components), least-squares regression (projecting data onto orthogonal subspaces), quantum mechanics (orthogonal states are mutually exclusive), and numerical linear algebra (orthogonal transformations preserve lengths and angles).
Key Formula
u and v are orthogonal if and only if u · v = u1*v1 + u2*v2 + ... + un*vn = 0
LaTeX: \mathbf{u} \perp \mathbf{v} \iff \mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^{n} u_i v_i = 0
| Symbol | Meaning | Unit |
|---|---|---|
| \mathbf{u}, \mathbf{v} | Vectors in ℝⁿ | — |
| u_i, v_i | i-th components of u and v | — |
| \mathbf{u} \cdot \mathbf{v} | Dot product (inner product) | — |
Worked Example
Problem
Verify that u = (3, −4, 0) and v = (4, 3, 5) are orthogonal and find a third vector orthogonal to both.
Solution
Step 1: Compute u · v = (3)(4) + (−4)(3) + (0)(5) = 12 − 12 + 0 = 0. ✓ Orthogonal. Step 2: Find w = u × v (cross product): w₁ = (−4)(5) − (0)(3) = −20 w₂ = (0)(4) − (3)(5) = −15 w₃ = (3)(3) − (−4)(4) = 9 + 16 = 25 Step 3: Verify w · u = (−20)(3) + (−15)(−4) + (25)(0) = −60 + 60 = 0. ✓ Step 4: Verify w · v = (−20)(4) + (−15)(3) + (25)(5) = −80 − 45 + 125 = 0. ✓
Answer
u · v = 0, confirming orthogonality. A third orthogonal vector is w = (−20, −15, 25).
Orthogonality vs. Orthonormality
| Property | Orthogonal Set | Orthonormal Set |
|---|---|---|
| Pairwise dot product (i ≠ j) | uᵢ · uⱼ = 0 | uᵢ · uⱼ = 0 |
| Self dot product | uᵢ · uᵢ = any positive value | uᵢ · uᵢ = 1 |
| Vector length | Any non-zero length | Length = 1 (unit vector) |
| Matrix property | AᵀA = diagonal | AᵀA = I (identity) |
| Example | {(1,0), (0,2)} in ℝ² | {(1,0), (0,1)} in ℝ² |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Eigenvector
An eigenvector of a square matrix A is a non-zero vector v that, when multiplied by A, yields a scalar multiple of itself: Av = λv, where λ is the corresponding eigenvalue. Geometrically, eigenvectors are the special directions in space that a linear transformation leaves invariant — only stretching or compressing along those directions without rotation. Eigenvectors are the foundation of matrix diagonalisation, principal component analysis, Google's PageRank algorithm, and the description of quantum states in physics.
Vector Space
A vector space is a set of objects called vectors, together with two operations — vector addition and scalar multiplication — that satisfy eight axioms including closure, associativity, distributivity, and the existence of zero and inverse elements. Vector spaces provide the foundational framework for linear algebra, generalising the familiar geometry of arrows in 2D and 3D to abstract settings of any dimension. They are essential in physics, engineering, computer graphics, machine learning, and quantum mechanics, where states, signals, and transformations are all described as elements of appropriate vector spaces.
Basis (Linear Algebra)
A basis of a vector space is a set of linearly independent vectors that spans the entire space, meaning every vector in the space can be expressed as a unique linear combination of the basis vectors. The number of vectors in any basis of a finite-dimensional vector space is the same and equals the dimension of the space. Bases are fundamental in coordinate systems, data compression, Fourier analysis, and the numerical solution of linear systems.
From Greek "orthos" (straight, right) + "gonia" (angle), meaning "right angle". The term has been in use in geometry since antiquity and was formalised for abstract vector spaces by mathematicians including Hilbert and Schmidt in the early 20th century.