Example 19 For any two vectors 
aand
rb
r
, we always have |
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|
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a b
a
b
⋅
≤
r
r
r
r
  (Cauchy-
Schwartz inequality) Solution The inequality holds trivially when either 
0  or  
0
a
b
=
r=
r
r
r Actually, in such a
situation we have |
|
0
|
||
|
a b
a
b
⋅
=
=
r
r
r
r So, let us assume that |
|
0
|
|
a
b
≠
≠
r
r