13880_lecture notes 2.5 - 2.6
TRANSCRIPT
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
CARTESIAN VECTORS ANDTHEIR ADDITION & SUBTRACTION
In&Class !ti%ities:
• Reading 'ui(
• !ppliations ) Rele%ane
• ! *nit +etor
• 3& +etor -ers
• !dding +etors
• Conept 'ui(
• E/aples
• !ttention 'ui(
-oda0s 2beti%es:
Students 4ill be able to5
a6 Represent a 3& %etor in aCartesian oordinate s0ste.
b6 7ind the agnitude andoordinate angles o8 a 3&
%etor6 !dd %etors 98ores6 in 3&
spae
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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READING QUIZ
1. +etor algebra, as 4e are going to use it, is based on a ::::::::::: oordinate s0ste.
!6 Eulidean B6 "e8t&handed
C6 ;ree< 6 Right&handed E6 Eg0ptian
#. -he s0bols α, β, and γ designate the :::::::::: o8a 3& Cartesian %etor.
!6 *nit %etors B6 Coordinate diretion angles
C6 ;ree< soieties 6 =, Y and > oponents
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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APPLICATIONS
In this ase, the po4erpole has gu0 4ireshelping to
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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APPLICATIONS 9ontinued6
In the ase o8 this radio to4er, i8 0ou
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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CARTESIAN UNIT VECTORS
-he unit %etors in the Cartesiana/is s0ste are i , j , and k . -he0are unit %etors along the positi%e /,0, and ( a/es respeti%el0.
Charateristis o8 a unit %etor :
a6 Its agnitude is 1.
b6 It is diensionless 9has no units6.6 It points in the sae diretion as the
original %etor 9 A6.
7or a %etor A, 4ith a agnitude
o8 !, an unit %etor is de8ined asu A = A ) ! .
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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CARTESIAN VECTOR REPRESENTATION
Consider a bo/ 4ith sides !=, !Y, and !> eters long.
-he %etor A an be de8ined as
A = (!= i + !Y j + !> k )
-he proetion o8 %etor A in the /&0 plane is !A. -he
agnitude o8 !A is 8ound b0 using the sae approah as a #& %etor5 !A 9!=
# !Y#61)# .
-he agnitude o8 the position %etor A an no4 be obtained as
! 99!A6# !>#6 ½ 9!=
# !Y# !>
#6 ½
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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*sing trigonoetr0, Ddiretion osines are 8ound using
-hese angles are not independent. -he0 ust satis80 the 8ollo4ing eFuation.
cos ² α + cos ² β + cos ² γ =
-his result an be deri%ed 8ro the de8inition o8 a oordinate diretion anglesand the unit %etor. Reall, the 8orula 8or 8inding the unit %etor o8 an0position %etor5
or 4ritten another 4a0, u A os α i + os β j + os γ k !
DIRECTION O" A CARTESIAN VECTOR
-hese angles are easured bet4een the %etorand the positi%e =, Y and > a/es, respeti%el0.-heir range o8 %alues are 8ro $G to 1$G
-he diretion or orientation o8 %etor A isde8ined b0 the angles ά, , and γ.
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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ADDITION O" CARTESIAN VECTORS (S#c$%o '!)
7or e/aple, i8
A != i !Y j !> k and
B B= i BY j B> k , then
A + B = 9!= B=6 i 9!Y BY6 j + 9!> B>6 k
or
A B = 9!= & B=6 i 9!Y & BY6 j + 9!> & B>6 k .
2ne indi%idual %etors are 4ritten in Cartesian 8or, it iseas0 to add or subtrat the. -he proess is essentiall0 thesae as 4hen #& %etors are added.
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
I*PORTANT NOTES
Soeties 3& %etor in8oration is gi%en as5
a6 @agnitude and the oordinate diretion angles, or,
b6 @agnitude and proetion angles.
You should be able to use both these sets o8in8oration to hange the representation o8 the%etor into the Cartesian 8or, i.e.,
F J1$ i #$ j 3$ k L M.
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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EA*PLE
16 *sing geoetr0 and trigonoetr0, 4rite F 1 and F 2 inCartesian %etor 8or.
#6 -hen add the t4o 8ores 9b0 adding / and 0&oponents6.
G
G%,#: -4o 8ores F 1 and F 2 are
applied to a hoo
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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7( N$$ 93)N6 3$$ M
7/ $ $ M
70 N$$ 9O)N6 O$$ M
Mo4, 4rite F 1 in Cartesian %etor 8or
9dont 8orget the units6.
F 1 J$ i O$$ j 3$$ k L M
Solution5
7irst, resol%e 8ore F 1.
EA*PLE 9ontinued6
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
Mo4, resol%e 8ore F 2.
7#( &$$ sin ONG − NQN. M
7# $$ os ONG NQN. M
7# an be 8urther resol%ed as,
7#/ NQN. os 3$G O. M
7#0 NQN. sin 3$G ##. M
-hus, 4e an 4rite5
F 2 JO. i ##. j − NQN. k L M
F2’
F2z
EA*PLE 9ontinued6
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
So F R F 1 F 2 and
F 1 J$ i O$$ j 3$$ k L M
F 2 JO. i ##. j − NQN. k L M
F R J O$ i Q3 j − #QQ k L M
EA*PLE 9ontinued6
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
CONCEPT QUESTIONS
1. I8 0ou 6 6 !ll o8 the abo%e.
#. 7or a 8ore %etor, the 8ollo4ing paraeters are randol0generated. -he agnitude is $. M, α 3$U , $U , V 1$$U. What is 4rong 4ith this 3& %etor ?
!6 @agnitude is too sall.
B6 !ngles are too large.
C6 !ll three angles are arbitraril0 pi
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
GROUP PROBLE* SOLVING
16 *sing the geoetr0 and trigonoetr0, resol%e, and 4rite
F 1 and F 2 in the Cartesian %etor 8or.#6 !dd F 1 and F 2 to get F R.
36 eterine the agnitude and angles α, β, γ .
G%,#: -he sre4 e0e is subeted
to t4o 8ores, F 1 and F 2.
"%-: -he agnitude and theoordinate diretion angleso8 the resultant 8ore.
P./:
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
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GROUP PROBLE* SOLVING 9ontinued6
7A an be 8urther resol%ed as,
71/ #$O. sin #NG Q.Q M
710 #$O. os #NG 1N.Q M
7irst resol%e the 8ore F 1.
71( & #N$ sin 3NG & 1O3.O M
7A #N$ os 3NG #$O. M
Mo4 4e an 4rite5F 1 JQ.Q i 1N.Q j − 1O3.O k L M
F´
F1z
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
GROUP PROBLE* SOLVING 9ontinued6
F 2 J $$ i ##. j #$$ k L M
-he 8ore F 2 an be represented in the Cartesian %etor 8or as5
F 2 O$$J os 1#$G i os ONG j os Q$G k L M
J $$ i ##. j #$$ k L M
Mo4, resol%e 8ore F 2.
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
GROUP PROBLE* SOLVING 9ontinued6
Mo4 8ind the agnitude and diretion angles 8or the %etor.
7R J9&113.O6# OQ.O# NQ.Q#L1)# ON.# ON M
α os&1 97R/
) 7R
6 os&1 9&113.O ) ON.#6 1$3.NG
β os&1 97R0 ) 7R6 os&1 9OQ.O ) ON.#6 1N.1G
γ os&1 97R( ) 7R6 os&1 9NQ.Q ) ON.#6 3.3G
So F R F 1 F 2 and
F 1 J Q.Q i 1N.Q j − 1O3.O k L M
F 2 J $$ i ##. j #$$ k L M
F R J &113.O i OQ.O j NQ.Q k L M
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
ATTENTION QUIZ
1. What is not true about an unit %etor, e.g., u A0
!6 It is diensionless. B6 Its agnitude is one.
C6 It al4a0s points in the diretion o8 positi%e =& a/is.
6 It al4a0s points in the diretion o8 %etor A.
#. I8 F J1$ i 1$ j 1$ k L M and
G J#$ i #$ j #$ k L M, then F G J :::: L N
!6 1$ i 1$ j 1$ k
B6 3$ i #$ j 3$ k
C6 1$ i 1$ j 1$ k
6 3$ i 3$ j 3$ k
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Mechanics for Engineers: Statics, 13th SI EditionR C Hibbeler and Kai Beng Yap
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End of the Lecture
Let Learning Continue