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

232 Chapter 7 Vectors
7.2.7 Orthogonalvectors
Iftheanglebetweentwovectorsaandbis90 ◦ ,thatisaandbareperpendicular,thena
and baresaidtobeorthogonal.
EXERCISES7.2
1 ForthearbitrarypointsA,B,C,DandE,findasingle
vector which isequivalent to
→
(a) DC + CB
→ →
(b) CE + DC
→
2 Figure7.18 shows acube. Letp = → AB,q= → AD and
r = → AE.Expressthe vectors representing → BD, → ACand
→
AGin terms of p,qand r.
F
G
3 Inatriangle ABC,Mis the midpoint ofBC,andNis
→
the midpoint ofAC.Show thatNM = 1 →
2AB.
4 Considerarectangle with verticesat E,F, GandH.
SupposeEF=pand → FG → = q.Expresseachofthe
vectorsEH,
→ → →
GH, FH andGE → in terms of p andq.
5 Ifaisan arbitraryvector, represent on adiagramthe
vectorsa, 1 2 a, −a ,3a, −3aand â.
4
B
p
E
r
A q
Figure7.18
C
D
H
6 Aparticleis positionedat the origin.Two forcesact
onthe particle.Thefirstforce has magnitude 7 N
andactsin the negativexdirection.The secondforce
has magnitude 12 Nandactsin theydirection.
Calculate the magnitude anddirectionofthe resultant
force.
7 Aforce of15Nacts atanangleof65 ◦ to thexaxis.
Resolvethisforce intotwo forces, onedirected along
thexaxisandthe other directed along the
yaxis.
Solutions
1 (a)
→
DB
(b)
→
DE
a
2 q−p,q+p,q+r+p
4 q,−p,q−p,−p−q
5 SeeFigureS.16.
6 13.9 Natan angleof59.7 ◦ to the negativexaxis
7 6.34 N, 13.59N
1– a
2
– –– a
4
FigureS.16
3a
–3a
7.3 CARTESIANCOMPONENTS
Consider thex--y plane in Figure 7.19. The general point P has coordinates (x,y). We
can join the origin to P by a vectorOP, → which is called the position vector of P, which
weoftendenotebyr.Themodulusofris |r| =r,andisthelengthofOP.Itispossible
→