Chapter 1.2, Problem 59EQ

Let u and v be non-zero vectors in 3-space. Prove that u and v are perpendicular if and only if |u × v| = |u||v|.

The Cauchy-Schwarz inequality says that if a = (a1,..., an) and b = (b1,..., bn) are two vectors in R", then lā · b| < |||||. In this exercise you will give a proof of this inequality using multivariable calculus. (a) Assume that the inequality is true for all 6 e R" with ||b|| the inequality must then be true for all be R". 1. Deduce from this that

Show that ||v||≤||v||₁||v|| about 1¹, 1², and 1 vector norms.