Chapter 10.3, Problem 51E

The Parallelogram Law states that

${\left|a+b\right|}^2+{\left|a-b\right|}^2=2{\left|a\right|}^2+2{\left|b\right|}^2$

1. (a) Give a geometric interpretation of the Parallelogram Law.
2. (b) Prove the Parallelogram Law. (See the hint in Exercise 50.)

The Parallelogram Law states that

${\left|a+b\right|}^2+{\left|a-b\right|}^2=2{\left|a\right|}^2+2{\left|b\right|}^2$

(a) Give a geometric interpretation of the Parallelogram Law.

(b) Prove the Parallelogram Law. (See the hint in Exercise 50.)

50. The Triangle Inequality for vectors is

$\left|a+b\right|⩽\left|a\right|+\left|b\right|$

1. (a) Give a geometric interpretation of the Triangle Inequality.
2. (b) Use the Cauchy-Schwarz Inequality from Exercise 49 to prove the Triangle Inequality. [Hint: Use the fact that |a + b|² = (a + b) · (a + b) and use Property 3 of the dot product.]