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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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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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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APP!ICATION" ;ontinued2

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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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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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U"ING T(E DOT PRODUCT TO DETER)INET(E ANG!E *ET+EEN T+O ,ECTOR"

≤ θ ≤ 1?$> .

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DETER)INING T(E PROECTION O# A ,ECTOR

"te.s:

1.

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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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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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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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u AO = r AO +r!J = ;−1+32 i @ ;#+32 j @ ;−#+32 k

Jr

E/A)P!E ;ontinued2

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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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GROUP PRO*!E) "O!,ING

Plan:

1.

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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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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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θ = os-1;F • r OA2+;< rJ!2

θ = os-11GO.O +;3$$ $.L$2 = 3L.1Q

Jr

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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

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13885_Lecture Notes 2.9