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which is the desired result. Note that the equality holds if and only if the two
vectors are linearly dependent (i.e., one vector is equal to a scalar multiplied
by the other vector).

Example 7.7
Show that for any three non-zero numbers, u , u , and u , the following ine-
2
1
3
quality always holds:
 1 1 1 
9 ≤ (u + u + u 3)  + +  (7.37)
1 2
 u 1 u 2 u 3 
PROOF Choose the vectors v and w such that:

/
v = u 1/2 u , / 12 u , 12 (7.38)
1 2 3

w =  1 1   12/ ,  1 2   12/ ,  1 3   12/ (7.39)



u 

u 

u 

then:
vw = 3 (7.40)
vv = ( u + u + u ) (7.41)
1 2 3

1
ww =  1 1 + u 1 2 + u    (7.42)


u
3
Applying the Cauchy-Schwartz inequality in Eq. (7.36) establishes the
desired result. The above inequality can be trivially generalized to n-ele-
ments, which leads to the following important result for the equivalent resis-
tance for resistors all in series or all in parallel.

Application
The equivalent resistance of n-resistors all in series and the equivalent resis-
tance of the same n-resistors all in parallel obey the relation:

n ≤ R series (7.43)
2
R
parallel

© 2001 by CRC Press LLC