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Scalarproductproperties:
abba-scalarproductiscommutative,
|=|
m
(
ab
|
)(
=
m
ab
)
|=
(
m
ba,
)
|
(
abcacbc
+
)
|=|+|
-thelawofdistributivemultiplicationover
addition.
Thedefinitionofscalarproductentailsthepropertyoforthog-
onalityoftwonon-zerovectorsaandbandtheformulaforthe
lengthofthevectora.Namely,twovectorsareorthogonal(the
anglebetweenthevectorsisright)iftheirscalarproductisequal
tozero,i.e.
ab
|=
0
.
Formulaforthelengthofthevectora:
a
=
aa
|=
a
2
=
a
1
2
+
a
2
2
+
a
3
2
.
(1.10)
Inarectangularcoordinatesystem,forthegiventwovectors
a
=
aaa
1
2
,]
3
and
b
=
bbb
1
2
,]
3
,thescalarproductinisexpressed
[,
[,
intheindexnotationintheformula:
ab
|=
δ
iji
ab
j
=
δ
i
1
ab
i
1
+
δ
i
2
ab
i
2
+
δ
i
3
ab
i
3
=
=
δ
1111
ab
+
δ
2222
ab
+
δ
3333
ab
=
=
ab
11
+
ab
22
+
ab
33
=
ab
ii
=
ab
kk
.
(1.11)
Intheabovecaseofscalarmultiplication,theKroneckerdelta
symbol
δ
ij
isused.Accordingtoitsdefinition,wehave:
δ
ij
={
[
[
0,gdy
1,gdy
i
i
#
=
j
j
.
,
Assumingorthogonalityofthecoordinatesystem,weobtain
ee
i
|
j
=
δ
ij
.
Ifaandbarenon-zerovectors,thenusingthedefinitionofscalar
product,theanglebetweenthevectorsiscalculatedusingthe
formula:
cos
u
=
a
1
2
+
ab
a
11
2
2
+
+
a
ab
3
22
2
b
+
1
2
ab
+
33
b
2
2
+
b
3
2
.
(1.12)
CHAPTER1
|
FUNDAMENTALSOFVECTORANDTENSORCALCULUS
———
———
———
——
21
