082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

9.7 The exponential form of a complex number 337
y
B
z 2
OC
z 1
BA
A
C
y
5
4
3
2
1
B
z 2
z 1 + z 2
z 1 – z 2
z 1
A
C
O
x
O 1
2 3 4 5 6 7 8 9 10
x
Figure9.8
Vector addition and subtraction.
Figure9.9
Diagram forExample9.15.
More generally, if z 1
and z 2
are any complex numbers represented on an Argand
diagram by the vectors OA ⃗ and OB ⃗ (Figure 9.8) then upon completing the parallelogram
OBCA, the sum z 1
+ z 2
is represented by the vector OC. ⃗ We can also obtain
a representation of the difference of two complex numbers in the following way. If
wewrite
z 3
=z 1
−z 2
=z 1
+(−z 2
)
andnotethatifz 2
isrepresentedbythevectorOB,then ⃗ −z 2
isrepresentedbythevector
−OB ⃗ = BO ⃗ = CA. ⃗ The complex number z 1
is represented by OA ⃗ = BC, ⃗ so that
z 1
+ (−z 2
) = BC ⃗ + CA ⃗ = BA. ⃗ Thus the difference z 1
− z 2
is represented by the
diagonal BA. To summarize, the sum and difference of z 1
and z 2
are represented by
the two diagonals of the parallelogram OBCA.
Example9.15 Represent z 1
= 6 + j and z 2
= 3 + 4j, and their sum and difference, on an Argand
diagram.
Solution WedrawvectorsOAand ⃗ OBrepresentingz ⃗ 1
andz 2
,respectively (Figure9.9).Thenwe
completetheparallelogramOACBasshown.Thesumz 1
+z 2
isthenrepresentedby ( ) OC ⃗
andthedifferencez 1
−z 2
byBA.Itiseasytoseethatthevector ⃗ BA ⃗ 3
(
= ,thatisit
−3
representsthecomplexnumber3 −3j,whileOC ⃗ 9
= ,thatisitrepresents9 +5j,so
5)
thatz 1
+z 2
=9+5jandz 1
−z 2
=3−3j.
9.7 THEEXPONENTIALFORMOFACOMPLEXNUMBER
You will recall from Chapter 6 that many functions possess a power series expansion,
thatisthefunctioncanbeexpressedasthesumofasequenceoftermsinvolvinginteger
powers ofx. For example,
e x =1+x+ x2
2! + x3
3! +···