Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 11 / i / Solution

Put $\mu =|b{|}^2>0$ and $\lambda =a\cdot b$. Expanding the nonnegative squared Euclidean norm of $c=\mu a-\lambda b$ givesDividing by $\mu$ and taking nonnegative square roots proves the Cauchy-Schwarz inequality:Equality holds exactly when $c=0$, equivalently when the nonzero vectors $a,b$ are linearly dependent.