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A Visual Proof of the Cauchy-Schwarz Inequality in 2D
A Visual Proof of the Cauchy-Schwarz Inequality in 2D
This illustrates the Cauchy-Schwarz inequality in two dimensions, which states:
ax+by=++sin(θ)⟹a,b·x,y≤a,bx,y
2
a
2
b
2
x
2
y
The gray area on the left is . The same gray area in the right hand image is ++sin(θ)=〈a,b〉〈x,y〉sin(θ). For , , hence the inequality.
ax+by=|〈a,b〉·〈x,y〉|
2
a
2
b
2
x
2
y
0≤θ≤
π
2
0≤sin(θ)≤1