Linear Algebra
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Angles and Orthogonality

Given two nonzero vectors, the angle between them satisfies cos⁡θ=(u⋅v)/(∥u∥⋅∥v∥)\cos \theta = (\mathbf{u} \cdot \mathbf{v}) / (\|\mathbf{u}\| \cdot \|\mathbf{v}\|). The Cauchy-Schwarz inequality guarantees this ratio always lies in [−1,1][-1, 1], so cos⁡−1\cos^{-1} is well-defined. The angle is always in [0°,180°][0°, 180°].

The most important special case is θ=90°\theta = 90° — when cos⁡θ=0\cos \theta = 0, i.e., when u⋅v=0\mathbf{u} \cdot \mathbf{v} = 0. We say u\mathbf{u} and v\mathbf{v} are orthogonal and write u⊥v\mathbf{u} \perp \mathbf{v}. Orthogonality generalizes the familiar notion of perpendicularity to all dimensions.

Note: orthogonality includes the case where one of the vectors is zero, since 0⋅v=0\mathbf{0} \cdot \mathbf{v} = 0 for all v\mathbf{v}.

Formal View

Definition 6.5 — Angle Between Vectors
For nonzero u,v∈Rm\mathbf{u}, \mathbf{v} \in \mathbb{R}^m, the angle between them is
θ(u,v):=cos⁡−1 ⁣(u⋅v∥u∥ ∥v∥)∈[0,π].\theta(\mathbf{u}, \mathbf{v}) := \cos^{-1}\!\left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\,\|\mathbf{v}\|}\right) \in [0, \pi].
Definition 6.6 — Orthogonality
Vectors u,v∈Rm\mathbf{u}, \mathbf{v} \in \mathbb{R}^m are orthogonal, written u⊥v\mathbf{u} \perp \mathbf{v}, if u⋅v=0\mathbf{u} \cdot \mathbf{v} = 0.
Example 6.7
In R2\mathbb{R}^2: (1,1)⊥(1,−1)(1,1) \perp (1,-1) since (1)(1)+(1)(−1)=0(1)(1) + (1)(-1) = 0. In R3\mathbb{R}^3: (1,0,0)⊥(0,1,0)(1,0,0) \perp (0,1,0) and (0,0,1)(0,0,1) — the standard basis vectors are mutually orthogonal.

Interactive Visualization

Dot Product and Angle

Why This Matters

Orthogonality is the central organizing principle of applied linear algebra.

  • Fourier analysis decomposes signals into orthogonal sine and cosine waves — orthogonality ensures no frequency "contaminates" another
  • In statistics, uncorrelated random variables are nearly the same concept as orthogonal vectors
  • PCA (principal component analysis) finds orthogonal directions of maximum variance in high-dimensional data
  • Error-correcting codes exploit orthogonality: a transmitted signal lies in one subspace, and errors lie in the orthogonal complement

Quiz

Question 1

Which pair of vectors is orthogonal?

Question 2

The zero vector is orthogonal to every vector.

Common Mistakes

  • Thinking perpendicular means u+v=0\mathbf{u} + \mathbf{v} = 0 — orthogonality is about the dot product, not the sum.
  • Confusing negative dot product (obtuse angle) with orthogonal (zero dot product) — orthogonal means exactly zero.
  • Assuming u⊥v\mathbf{u} \perp \mathbf{v} and u⊥w\mathbf{u} \perp \mathbf{w} implies v⊥w\mathbf{v} \perp \mathbf{w} — orthogonality to a fixed vector does not force two vectors to be orthogonal to each other.