Exercise 62

The Triangle Inequality for vectors is \(|\mathbf{a} + \mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}|\) (a) Give a geometric interpretation of the Triangle Inequality. (b) Use the Cauchy-Schwarz Inequality from Exercise 61 to prove the Triangle Inequality. [Hint: Use the fact that \(|\mathbf{a} + \mathbf{b}|^2 = (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b})\) and use Property 3 of the dot product.]