082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

7.2 Vectors and scalars: basic concepts 225
A
a
B
A
C
AB
B
CD
D
Figure7.1
Avector, AB,oflength → 4.
Figure7.2
Two equal vectors.
a vector in the direction shown by the arrow on AB. This vector is denoted by AB.
→
→
Note that AB ≠ BA. → The vector AB → →
is directed from A to B, but BA is directed from
BtoA.
An alternative notation is frequently used: we denote AB → by a. This bold notation
is commonly used in textbooks but the notation a is preferable for handwritten work.
Wenowneedtorefertothediagramtoappreciate theintended direction.Thelengthof
the line segment represents the magnitude, or modulus, of the vector and we use the
notation | → AB|, |a| or simply a to denote this. Note that whereas a is a vector, |a| is a
scalar.
7.2.1 Negativevectors
The vector −a is a vector in the opposite direction to, but with the same magnitude as,
a. Geometrically itwill be → BA. Thus −aisthe same as → BA.
7.2.2 Equalvectors
Twovectorsareequaliftheyhavethesamemagnitudeanddirection.InFigure7.2vectors
→ CDand → ABareequaleventhoughtheirlocationsdiffer.Thisisausefulpropertyof
vectors:avectorcanbetranslated,maintainingitslengthanddirectionwithoutchanging
thevectoritself.Thereareexceptionstothisproperty.Forexample,weshallsoonmeet
position vectors which are used to locate specific fixed points in space. They clearly
cannot be moved around freely. Nevertheless most of the vectors we shall meet can be
translated freely, and assuch are often calledfree vectors.
7.2.3 Vectoraddition
Itisfrequentlyusefultoaddtwoormorevectorstogetherandtheadditionofvectorsis
definedbythetrianglelaw.ReferringtoFigure7.3,ifwewishtoaddABto → CD,
→ CDis
translateduntilCandBcoincide.Asmentionedearlier,thistranslationdoesnotchange
→
thevectorCD.Thenthesum,
→ AB+ CD,isdefinedbythevectorrepresentedbythethird
→
sideof the completed triangle, thatis → AD(Figure 7.4).Thus,
→
AB + → CD = → AD
→
Similarly,ifABisdenotedbya,
→ CDisdenotedbyb, ADisdenotedbyc,thenwehave
c=a+b
Wenotethattoaddaandbatriangleisformedusingaandbastwoofthesidesinsuch
a way that the head of one vector touches the tail of the other as shown in Figure 7.4.
The suma +bisthen represented by the third side.
→
→