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PROOF The proof is straightforward. Using Eq. (7.37) and recalling Ohm’s
law for n resistors {R , R , …, R }, the equivalent resistances for this combina-
2
n
1
tion, when all resistors are in series or are all in parallel, are given respec-
tively by:
R = R + R +…+ R (7.44)
series 1 2 n

and

1 1 1 1
= + +…+ (7.45)
R R R R
parallel 1 2 n
Question: Can you derive a similar theorem for capacitors all in series and all
in parallel? (Remember that the equivalent capacitance law is different for
capacitors than for resistors.)

7.4.2 Triangle Inequality
This is, as the name implies, a generalization of a theorem from Euclidean
geometry in 2-D that states that the length of one side of a triangle is smaller
or equal to the sum of the the other two sides. Its generalization is

uv+ ≤ u + v (7.46)

PROOF Using the relation between the norm and the dot product, we have:

uv+ 2 = uv uv+ + = u v + 2 u v + v v
(7.47)
2
2
2
= u + 2 u v + v ≤ u + 2 u v + v 2
Using the Cauchy-Schwartz inequality for the dot product appearing in the
previous inequality, we deduce that:

2
2
uv+ 2 ≤ u + 2 u v + v = ( u + ) 2 (7.48)
v
which establishes the theorem.

Homework Problems
Pb. 7.5 Using the Dirac notation, generalize to n-dimensions the 2-D geom-
etry Parallelogram theorem, which states that: The sum of the squares of the diag-
onals of a parallelogram is equal to twice the sum of the squares of the side; or that:

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