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VECTORANDTENSORCALCULUSFORENGINEERS
x
a
e
0
2
2
2
e
1
A(x
A,x
1
AB=a
A)
1
a
1
B(x
B,x
1
x
B)
1
1
x
a
x
a
z
β
x
z
β
z
a
β
y
a
y
y
figure1.5
18
Fromthedefinitionofthesumofvectors,theformulafollows:
3
a
=
a
11
e
+
a
22
e
+
a
33
e
=
a
1
+
a
2
+
a
3
=Σ
a.
i
i
=
1
Axisversors(orbasisvectors)canbepresentedasfollows:
e
1
=
[1,0,0]
,
e
2
=
[0,1,0]
,
e
3
=
[0,0,1]
.
(1.4)
Alternatively,theversorsofthecoordinatesystemwillalsobe
designatedasi,j,k.
Letthenon-zerovector
a
=
aaa
x
y
,]
z
formangles
BBB
x
y
,
z
[,
,
withthecoordinateaxes(Fig.1.5).Thecosinesoftheseangles
arethedirectionalcosinesofthevectora,whereas:
cos
B
x
=
a
a
x
,cos
B
y
=
a
a
y
,cos
B
z
=
a
a
z
.
(1.5)
Bysquaringbothsidesoftheequation(1.5)andaddingside-by-side,
weobtain:
cos
2
B
x
+
cos
2
B
y
+
cos
2
B
z
=.
1
(1.6)
Thesumordifferenceoftwovectors,
a
=
aaa
1
2
,]
3
and
[,
b
=
bbb
1
2
,]
3
,isformedbyaddingorsubtractingvectorco-
[,
ordinates:
