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Thesum
ab
+
ofvectorsaandbisvectorc,whichisformedas
follows:drawtheavecpointofvector
cab
=+
istheOpointits
terminalpointistheterminalpointofthevectorb(Fig.1.1).The
sumofthevectorsdoesnotdependontheselectionofpointO.
Thediference
ab
−
ofvectorsaandbisthesumofthevector
aandthevector−boppositetothevectorb.Thisdiferenceis
definedusingtheformula:
−b
b
c=(a+b)
(a−b)
a
(a−b)
aba
−=+−
(
b.
)
Thesumandthediferencecanberepresentedusingaparallel-
ogram,asshowninFig.1.1.
Theassociativelawisapplicabletovectoraddition.
Inaccordancewiththeassociativelaw,theresultingvectorr
(Fig.1.2),beingthesumofthreevectorsa,bandc,maybe
calculatedasfollows:
figure1.1
ra
=+
(
bc
+
)(
=
ab
+
)
+
c.
O
r
S
(a+b)
(b+c)
a
c
P
Q
b
Thecommutativelawisapplicablehere,i.e.theorderofvector
additionisfree,thatis:
cabba.
=+=+
Thepropertiesofthevectoradditioncanbesummarisedas
follows:
a
+
(
bc
+
)(
=
ab
+
)
+
c(commutativelaw),
abba(associativelaw),
a0a,
a
+=+
+=
+−
(
a
)
=
0.
Thesumofanynumberofvectorsmaybeobtainedusingthe
presentedmethod(Fig.1.3).Theclosingsideofthepolygonis
asumofvectors,howevertheorderofvectoradditionisfree.
Thesumvectorisformedbyconnectingtheinitialpointofthe
firstvectorwiththeterminalpointofthelastvector.
figure1.2
CHAPTER1
|
FUNDAMENTALSOFVECTORANDTENSORCALCULUS
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