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Cauchy-Schwarz Inequality Calculator

Online Cauchy-Schwarz verifier: compare the dot-product square with the product of squared magnitudes for two 2D vectors.

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(a₁b₁+a₂b₂)²≤(a₁²+a₂²)(b₁²+b₂²)
Components
Components
Components
Components

📖 Tutorial | Cauchy

1. Definition
For two 2D vectors, (a·b)²≤|a|²|b|², with equality when they are collinear (proportional).
2. Formula & Notation
a,bVectors
a·bDot product
3. Properties & Laws
  • (a·b)²≤|a|²|b|²;
  • |a·b|≤|a||b|;
  • Equality when collinear;
  • Extends to n dimensions.
4. Steps
  1. Enter four components;
  2. Click "Verify";
  3. Or click "Load Example".
5. Worked Examples
Example: a=(1,2), b=(3,4): dot=11, L=121, R=125.
6. Common Pitfalls
Do not drop squares;
Equality means proportional;
Dot may be negative, its square not.

❓ FAQ | Cauchy

Form?
Dot squared ≤ product.
Equality?
Collinear.
Vector form?
|a·b|≤|a||b|.
Uses?
Proofs, optimization.
Negative dot?
Square it.
nD?
Holds.
Cosine link?
|cosθ|≤1.