Cauchy Schwarz Strengthening with Scalar Multiples

Question

Cauchy-Schwarz inequality actually can be strengthened to include when is equality true. Try to show the following: Let vector x and vector y be two vectors in R^n, with y ≠ 0​. Then ∣ x ⋅ y ∣ = ∥ x ∣ ∥ y ∥ if and only if x  is a scalar multiple of y.

This equality case is often quite useful to show if two vectors are parallel to each other, i.e. both parallel to the same line.

Answer 1

Without loss of generality, assume |y| = 1. Then |x • y| = |x| so that

|x - (x • y)y|² = |x|² - 2x • (x • y)y + |(x • y)y|²

.....................= |x|² - 2(x • y)² + (x • y)² |y|²

.....................= |x|² - 2|x|² + |x|²

.....................= 0

Therefore x - (x • y)y = 0, i.e., x = (x • y)y. Hence, x is a scalar multiple of y.