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VECTORANDTENSORCALCULUSFORENGINEERS
D
A
4
j
a
i
figure1.8
M
c
b
3
B
C
N
Solution
AsthepointsMandNarecentresofthesidesoftherectangle
ABCD(Fig.1.8),therefore
where:i,j,karetheversorsofaxes.
Asassumed,thevectorcmustbethesumoftwovectors,i.e.
Bycomparingnowthecoordinatesatthesameversorsonboth
sidesoftheequation,weobtaintwoequations:
c
O
a
=
(
|
=
k
EXAMPLE1.3
IntherectangleABCD,MandNarethecentresofsides
directionsofvectorsAM=a
DC=,
O
3
2
3
2
a
i
i
+
+
+
4
3
4
B
j
j,
b,therefore:
N
|
)
BC=.DecomposethevectorAC=c
+
b
B
=
(
3
4
3
i
i
+
+
2
2
j,
j
)
c
=
=
ą
3
3
i
i
+
+
4
4
andAN=b
j.
j,
ą
ą
.
intothe
3
2
O
+
3
B
=,4
3
O
+
2
B
=.
4
Bysolvingthesystemofequations,weobtain:
O
=
B
=.
2
3
Answer
c
=
2
3
a
+
2
3
b.
1.5.Scalarproduct
Thescalarproducta×boftwonon-zerovectorsaandbisthe
numberequaltotheproductofthelengthsofthesevectorsand
thecosineoftheanglebetweenthem:
ab
|=
ab
cos
u
,where
u
=
<
(,)
ab
.
20
(1.9)
