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

7.3 Cartesian components 239
3 Ifa=4i−j+3kandb=−2i+2j−k,findunit
vectors in the directions ofa,b andb −a.
4 Ifa = 5i −2j,b = 3i −7jandc = −3i +4j,express,
in termsof iand j,
a+b a+c c−b 3c−4b
Drawdiagrams to illustrate your results.Repeat the
calculations usingcolumn vectornotation.
5 Writedown aunit vectorwhich is parallel to the line
y=7x−3.
6 Find → PQwhere Pisthe point in three-dimensional
space with coordinates (4,1,3) andQisthe point
with coordinates (1,2,4).Find the distancebetween
PandQ. Further,findthe position vectorofthe point
dividing PQ in the ratio 1:3.
7 If P,QandRhave coordinates (3,2,1), (2,1,2) and
(1,3,3),respectively, use vectors to determine which
pair ofpointsare closestto each other.
8 Considerthe robot ofExample7.4. Thelinkvectors
have the following values:
a =12i+18j+k
b=6i−3j+8k
c =3i+2j−4k
d = 0.5i −0.2j +0.6k
Calculate the length ofeach ofthe links and the
position vector ofthe tip ofthe robot.
9 Show that the vectorsa = i +j andb = −i +jare
linearly independent.
10 Show that the vectorsi,j and kare linearly
independent.
Solutions
1
→
PQ=−6i+3j | → PQ| = 6.71
2 3i+4j,i−5j,−2i−9j
−1
unitvector: √ (2i +9j)
85
3
1
√ (4i −j +3k), 1 (−2i +2j −k),
26 3
1
√
61
(−6i +3j −4k)
4 8i −9j,2i +2j, −6i +11j,−21i +40j
5
6
1
√
50
(i +7j)
→
PQ = −3i +j+k,distancefrom Pto Q = 3.32,
1
(13i +5j +13k)
4
7 PandQ
8 21.66,10.44,5.39,0.81,21.5i +16.8j +5.6k
TechnicalComputingExercises7.3
To plotadisplacement vectorbetween points(1,2)and
(3,4)in MATLAB ® wecouldtype:
X=[1 3];
Y=[2 4];
plot(X,Y)
whichwould resultin aplotcontaining asingle line
between the two points.Notice the order in whichthe
coordinatesare passed to the plotfunction. Thefirst
entry in Xandthe firstentry in Yare forthe first
location andsoon. Ifwewantedto plotasecond
displacement vector from (3,4)to (5,6)wewould
type:
X=[1 3 5];
Y=[2 4 6];
plot(X,Y)
Now consider the problemofroutingan automated
vehicle between the following positionson afactory
floor:
Doorway: A =(0,0)
Resistorbin: B =(1,1)
Capacitorbin: C = (1,-0.5)
PCBrack: D =(2,0)
Assemblyline: E =(2.5, 0)
Forsimplicityyou mayassume that the robotwill
notcollidewith any objects ifittravels directly
betweenany ofthose two points.
1 Plotthe displacement vectors forthe robotifit
enters through the doorway andvisits the resistor
bin,the PCBrackandthenthe assemblyline. How
would thisactionbe writtenin vectornotation?