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O
e
s
a
I
a
s
s
ab
+=
[
a
1
+
ba
1
2
+
ba
2
,
3
+
b
3
](
=
a
i
+
b
i
)
e,
i
,
(1.7a)
a
+−
(
b
)[
=
a
1
−
ba
1
2
−
ba
2
,
3
−
b
3
](
=
a
i
−
b
i
)
e.
i
,
(1.7b)
Theprojection
a
s
ofthevectoraontheaxisOsisavector
(Fig.1.6),theinitialpointofwhichistheprojectionofthestarting
pointandtheterminalpointistheprojectionoftheterminal
pointofthevectoraonthisaxis.Thecoordinate
a
s
ofthevector
arelativetotheaxis
Os
istheprojectionmeasure(orcoordinate)
ofthevector
a
s
relativetothisaxis.Assumingthat
e
s
isthe
versoroftheaxisOs,weobtain:
figure1.6
x
x
y
O
O
z
O
x
x
z
O
figure1.7
y
y
y
z
z
a
s
=a
cos
I
,
a
s
=
a
ss
e.
(1.8)
Wewillrelateallcalculationstotheright-handedcoordinate
system(Fig.1.7).
EXAMPLE1.2
Threevectorsaregiven:
a
=
[
1,1,
−
]
b
=
[
4,3,
]
c
=−
[
10,11.
−
]
Presentthevectorcasalinearcombinationofvectorsaandb.
Solution
Letuswritethevectorinthefollowingform:
c
=
O
a
+
B
b
.
Bymultiplyingbothsidesofthevectorequationbyversorsiandj,
weobtaintwolinearequations:
O
a
x
+
B
b
x
=,
c
x
O
a
y
+
B
b
y
=
c
y
,
therefore
O
+
4
B
=−
10
,
−+
O
3
B
=−.
11
Bysolvingthesystemofequations,weobtain:
O
=
2
i
B
=−
3
.
Answer
c
=
2
a
−
3
b.
CHAPTER1
|
FUNDAMENTALSOFVECTORANDTENSORCALCULUS
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