The Schwarz inequality upsilon middot w lessthanorequalto up

The Schwarz inequality |upsilon middot w| lessthanorequalto ||upsilon|| ||w|| by algebra instead of trigonometry: (a) Multiply out both sides of (upsilon_1 w_1 + upsilon_2 w_2)^2 lessthanorequalto (upsilon_1^2 + upsilon_2^2)(w_1^2 + w_2^2). (b) Show that the difference between those two sides equals (upsilon_1 w_2 - upsilon_2 w_1)^2. This cannot be negative since it is a square-so the inequality is true.

Solution

We have (v₁w₁ +v₂ w₂)² = v₁²w₁²+v₂²w₂²+2v₁w₁v₂w₂ ; and (v₁²+v₂²)(w₁²+w₂²) = v₁²w₁²+v₂²w₂²+v₁²w₂² + v₂²w₁² so that (v₁²+v₂²)(w₁²+w₂²) - (v₁w₁ +v₂ w₂)² = v₁²w₁²+v₂²w₂²+v₁²w₂² + v₂²w₁² – (v₁²w₁²+v₂²w₂²+2v₁w₁v₂w₂) =           v₁²w₂² + v₂²w₁² -2v₁w₁v₂w₂ = (v₁ w₂ – v₂ w₁)² . Now, (v₁ w₂ – v₂ w₁)² being a square, cannot be negative. Therefore, (v₁²+v₂²)(w₁²+w₂²)-(v₁w₁+v₂ w₂)² 0 or, (v₁²+v₂²)(w₁²+w₂²) (v₁w₁ +v₂ w₂)² or, (v₁w₁ +v₂ w₂)² (v₁²+v₂²)(w₁²+w₂²)