Creacion De Un Programa De Red

The Cauchy-Schwarz Inequality and the Triangle Inequality
The Cauchy-Schwarz inequality and the triangle inequality are important techicalinequalities that have
widespread applications, both theoretical and practical. (Indeed, as you will see below, the Cauchy Schwarz
Inequality is crucialfor proving the Triangle Inequality.)
Theorem 1 (Cauchy-Schwarz Inequality). For any vectors ~u and ~v, both in R2
or both in R3
,
j~u
~vj k~uk k~vk: (1)
Proof . If ~u = ~0 or ~v = ~0, then (1) just says 0  0, which is true. So suppose that ~u and ~v are nonzero
vectors. Then, asproved in class,
~u
~v = k~uk k~vk cos();
where  is the angle between the vectors. Taking absolute values then gives
j~u
~vj = k~uk k~vk jcos()j 
"
j cos()j1
k~uk k~vk:
Exercise 1. When will the inequality actually be an equality?
Theorem 2 (Triangle Inequality). For any vectors~u and ~v, both in R2
or both in R3
,
k~u + ~vk  k~uk + k~vk: (2)
Proof. Since both sides of (2) are nonnegative, (1) is equivalent to

k~u +~vk

2


k~uk + k~vk

2
;
this is what we will prove.

k~u + ~vk

2
= (~u + ~v)
(~u + ~v)
= ~u
(~u + ~v) + ~v
(~u + ~v)
= ~u
~u + ~u
~v + ~v
~u + ~v
~v
= k~uk
2
+ 2~u
~v + k~vk
2
 k~uk
2
+ 2j~u
~vj + k~vk
2
by Cauchy-Schwarz !  k~uk
2
+ 2k~uk k~vk +k~vk
2
=

k~uk + k~vk

2
:
Exercise 2. What does the Triangle Inequality have to do with triangles? (Draw a picture of ~u, ~v, and
~u + ~v.)