Cauchy–Schwarz inequality intuition

Question

We learned in class about the Cauchy–Schwarz inequality. Can some one please give me a geometric intuition or any intuition in general for why this inequality is true? Thank you.

Answer 1

If $\vec u,\vec v$ are two vectors then intuitively the dot product $<\vec u,\vec v>$ of them is the product of their lengths times the $\cos$ of the "angle" between $\vec u, \vec v$. The C-S inequality says that $$|<\vec u,\vec v>|\le ||\vec u||\cdot ||\vec v||.$$ So the geometric meaning of the inequality is that $|\cos|\le 1$. In fact you can define the "angle" between $\vec u,\vec v$ as the $\cos^{-1}$ of $$\frac{|<\vec u,\vec v>|}{||\vec u||\cdot ||\vec v||}.$$ If the vectors have many coordinates then this $\cos$ is called the "correlation". It is the statistical intuition for C-S.

Answer 2

Here's a way to put it: $$\prod{\sum} \ge \sum{\prod}.$$

The product of the sum of squares is greater than or equal to the square of the sum of products. It's an interesting intuition that you can put thought into. I'll leave you to it to analyze.

It also generally makes sense, because if you observe the behavior of numbers multiplying gets you bigger than adding. (Obviously not for numbers less than 1.) The LHS describes a product and the RHS describes a sum. Again, these are obviously observations and are not always true, nor are a proof.

-FruDe