11. Cauchy–Schwarz Inequality

Where:

• : dot product of vectors and
• and : magnitudes (lengths) of vectors and

Step 1: Understand the dot product

The dot product of two vectors and is calculated as:

The magnitude (or norm) of a vector is:

Step 2: Define a new function

We introduce a parameter and define a new vector:

Now, consider the dot product of this new vector with itself, which is always non-negative because it represents the square of the magnitude of :

This inequality makes sense because the dot product of any vector with itself is the square of its magnitude, and a square is always non-negative.

Step 3: Expand the dot product

Expand :

Now, we apply the distributive property of the dot product, which behaves similarly to the distributive property of multiplication. We expand each term:

Since the dot product is commutative (), we can rewrite this as:

This simplifies to:

We’ve now expressed the result of expanding the dot product as a quadratic expression in , where

• is a constant term,
• is the linear term in
• is the quadratic term

Step 4: Treat as a quadratic equation

Now that we have the quadratic expression:

We recognize this as a standard quadratic inequality of the form , where:

For any quadratic expression to always be non-negative, its discriminant must be less than or equal to zero. The discriminant of a quadratic equation is given by:

Substituting in the values of , , and from our expression:

Simplifying:

Step 5: Apply the discriminant condition

For the quadratic inequality to hold, the discriminant must be less than or equal to zero:

Divide by 4:

Take the square root of both sides: