The Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality is a fundamental principle in mathematics that relates two sets of numbers or vectors. It states that the absolute value of their dot product (a measure of how much they point in the same direction) is always less than or equal to the product of their lengths (magnitudes). Think of it as measuring how aligned two directions are: the more they point the same way, the closer the dot product gets to this maximum; if they are orthogonal (perpendicular), the dot product is zero. It helps ensure relationships between quantities remain balanced and consistent.