WOLFRAM|DEMONSTRATIONS PROJECT

A Visual Proof of the Cauchy-Schwarz Inequality in 2D

​
|a|
|b|
This illustrates the Cauchy-Schwarz inequality in two dimensions, which states:
ax+by=
2
a
+
2
b
2
x
+
2
y
sin(θ)⟹a,b·x,y≤a,bx,y
.
The gray area on the left is
ax+by=|〈a,b〉·〈x,y〉|
. The same gray area in the right hand image is
2
a
+
2
b
2
x
+
2
y
sin(θ)=〈a,b〉〈x,y〉sin(θ)
. For
0≤θ≤
π
2
,
0≤sin(θ)≤1
, hence the inequality.