Cauchy–Schwarz

Dot products are bounded

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Cauchy–Schwarz: $(\mathbf u \cdot \mathbf v)^2 \le |\mathbf u|^2 |\mathbf v|^2$.
• It bounds expressions like 3x + 4y on a circle.
• Correlation coefficients are cosines, so they lie between −1 and 1.

Steps

1. Vectors and cosines
2. Maximise on a circle
3. In three dimensions
4. Closest point to a line
5. Around the circle
6. Where it shows up