13885_Lecture Notes 2.9
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Transcript of 13885_Lecture Notes 2.9
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8/18/2019 13885_Lecture Notes 2.9
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8/18/2019 13885_Lecture Notes 2.9
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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.
DOT PRODUCT
In-Class Activities:
• Che& Ho'e(or&
• Reading )ui*
• !ppliations+Rele%ane
• ot produt - einition
• !ngle eter'ination
• eter'ining the
Pro/etion• Conept )ui*
• 0roup Proble' Sol%ing
• !ttention )ui*
Today’s Objective:
Students (ill be able to use the %etordot produt to
a2 deter'ine an angle bet(eent(o %etors, and
b2 deter'ine the pro/etion o a %etor
along a speiied line.
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8/18/2019 13885_Lecture Notes 2.9
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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.
READING QUI
1. he dot produt o t(o %etors P and Q is deined as
!2 P ) sin θ B2 P ) os θ
C2 P ) tan θ 2 P ) se θ
#. he dot produt o t(o %etors results in a 444444444 5uantit6.
!2 Salar B2 7etor C2 Co'ple8 2 9ero
P
Q
θ
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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.
APP!ICATION"
I 6ou &no( the ph6sialloations o the our ableends, ho( ould 6oualulate the angle bet(eenthe ables at the o''on
anhor:
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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.
APP!ICATION" ;ontinued2
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
DE#INITION
he dot produt o %etors A and B is deined as A$B = ! Bos θ.
he angle θ is the s'allest angle bet(een the t(o %etors andis al(a6s in a range o $> to 1?$>.
Dot P%od&ct C'a%acte%istics:
1. he result o the dot produt is a salar ;a positi%e ornegati%e nu'ber2.
#.he units o the dot produt (ill be the produt o theunits o the A and B %etors.
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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.
DOT PRODUCT DE#INITON;ontinued2
E8a'plesB6 deinition, i $ j = $
i $ i = 1
A • B = ;!8 i @ !6 j @ !* k 2 $ ;B8 i @ B6 j @ B* k 2
= !8 B8 @ !6B6 @ !*B*
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8/18/2019 13885_Lecture Notes 2.9
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
U"ING T(E DOT PRODUCT TO DETER)INET(E ANG!E *ET+EEN T+O ,ECTOR"
≤ θ ≤ 1?$> .
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8/18/2019 13885_Lecture Notes 2.9
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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.
DETER)INING T(E PROECTION O# A ,ECTOR
"te.s:
1.
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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.
3. I needed, the pro/etion an be (ritten as a %etor, A|| ,
b6 using the unit %etor uaa´ and the 'agnitude ound in
step #. A|| = !FF uaa´
G. he salar and %etor or's o the perpendiularo'ponent an easil6 be obtained b6
! ⊥ = ;!# - !FF
#2 and
A = A – A||;rearranging the %etor su' o A = A
+ A|| 2
DETER)INING T(E PROECTION O# A ,ECTOR;ontinued2
´
´
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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.
E/A)P!E
Plan:
1. hoo& at point !.#ind: he angle bet(een the
ore %etor and the line!J, and the 'agnitude othe pro/etion o the ore
along the line !J.
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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.
E/A)P!E ;ontinued2
r AO = −1 i @ # j − # k 'r!J = ;-12
# @ ## @ ;-#2#1+# = 3 '
F = − L i @ M j @ 3 k &N
< = ;-L2# @ M# @ 3#1+# = 11.## &N
θ = os-1;F • r AO2+;< r!J2
θ = os-1 1? + ;11.## × 32 = O.LQ
F • r AO = ;− L2;−12 @ ;M2;#2 @ ;32;−#2 = 1? &N '
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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.
u AO = r AO +r!J = ;−1+32 i @ ;#+32 j @ ;−#+32 k
Jr
E/A)P!E ;ontinued2
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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 QUI
1. I a dot produt o t(o non-*ero %etors is $, thenthe t(o %etors 'ust be 4444444444444 to eahother.
!2 Parallel ;pointing in the sa'e diretion2
B2 Parallel ;pointing in the opposite diretion2
C2 Perpendiular
2 Cannot be deter'ined.
#. I a dot produt o t(o non-*ero %etors e5uals -1, thenthe %etors 'ust be 44444444 to eah other.
!2 Collinear but pointing in the opposite diretion
B2 Parallel ;pointing in the opposite diretion2 C2 Perpendiular
2 Cannot be deter'ined.
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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 PRO*!E) "O!,ING
Plan:
1.
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
GROUP PRO*!E) "O!,ING ;ontinued2
r OA = −$.GO$ i @ $.3$$ j @ $.#L$ k '
rJ! = ;−$.GO$2# @ $.3$$# @ $.#L$# 1+# = $.L$ '
uOA = r OA + rJ! = -$.O i @ $.O$ j @ $.G33 k
F´
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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap
© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.
GROUP PRO*!E) "O!,ING ;ontinued2
F = −1O$ sin 3$Qi @ 3$$ os 3$Q j @ 1O$ os 3$Qk N
F = − O i @ #OM.? j @ 1#M.M k N
< = ;-O2# @ #OM.?# @ 1#M.M#1+# = 3$$ N
F´
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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 PRO*!E) "O!,ING ;ontinued2
θ = os-1;F • r OA2+;< rJ!2
θ = os-11GO.O +;3$$ $.L$2 = 3L.1Q
Jr
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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 QUI
1. he dot produt an be used to ind all o the ollo(ing
e8ept 4444 . !2 su' o t(o %etors
B2 angle bet(een t(o %etors
C2 o'ponent o a %etor parallel to another line
2 o'ponent o a %etor perpendiular to another line
#.
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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
End of the Lecture
Let Learning Continue