Proving Inequalities: Interactive proof of Cauchy–Schwarz, the triangle inequality and the absolute value triangle inequality, with adjustable angle and vector length

To prove one side is bigger, never measure either side: measure the gap

Cauchy–Schwarz, the triangle inequality and |a + b| ≤ |a| + |b| all come from one move: write the difference between the two sides as something that cannot be negative, then read the inequality off it. Here ‖u‖ is fixed at 1. Change the angle between the vectors and the length of v, and watch what the gap between the two sides does as the shaded area grows and shrinks.

Gap between the two sides gap 0.50
Change the angle. Watch where the projection lands and what the shaded area does.
Which inequality
|u·v| 0.50 ≤ 1.00 = ‖u‖‖v‖ · gap 0.50

‖u‖²‖v‖² − (u·v)² = ‖u‖²‖v‖² sin²θ = area² ≥ 0, so (u·v)² ≤ ‖u‖²‖v‖² and |u·v| ≤ ‖u‖‖v‖.

Angle θ between u and v60°
0°45°90°135°180°
Length of v (‖u‖ = 1)1.00
0.51.01.52.0
|u·v|
0.50
‖u‖‖v‖
1.00
Which proof technique

Know This

To prove A ≥ B, show A − B is a perfect square, a squared area or some other quantity that is never negative.