ENGI 1313 Mechanics I Lecture 07: Vector Dot Product Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland [email protected].

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Transcript ENGI 1313 Mechanics I Lecture 07: Vector Dot Product Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland [email protected]. 

ENGI 1313 Mechanics I
Lecture 07:
Vector Dot Product
Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]
Chapter 2 Objectives
to review concepts from linear algebra
 to sum forces, determine force resultants
and resolve force components for 2D
vectors using Parallelogram Law
 to express force and position in Cartesian
vector form
 to examine the concept of dot product

2
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07
Lecture 07 Objectives

3
to examine the concept of dot product
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07
Overview of Dot Product

Definition
 
 
A  B  A B cos 

0    180

Laws of Operations

Commutative law
 
 
A B  B  A

Scalar Multiplication
 
 
 


c A B  A B c  c A B  A  c B


 

Distributive law
 
 
 
 

A  B  C  A  B  A C

4
 
 
© 2007 S. Kenny, Ph.D., P.Eng.
 

ENGI 1313 Statics I – Lecture 07

Overview of Dot Product (cont.)

Dot Product of Cartesian Vectors
 
A B




A x ˆi 
 
Ax B x
 
Ax By
 
Ay Bz







ˆ
ˆ
ˆ
ˆ
A y j  A z k  B x i  B y j  B z kˆ
 
 
ˆi  ˆi  A B ˆj  ˆj  A B kˆ  kˆ 
y
y
z
z
 
 
ˆi  ˆj  A B ˆi  kˆ  A B ˆj  ˆi 
x
z
y
x
 
 
ˆj  kˆ  A B kˆ  ˆi  A B kˆ  ˆj
z
x
z
y
 
 
 
 
 
A  B  A B cos 
 
A B 
 
 
 

 
 
 







A x ˆi  A y ˆj  A z kˆ  B x ˆi  B y ˆj  B z kˆ
ˆi  ˆi  ˆj  ˆj  kˆ  kˆ  1 1  cos 0   1
ˆi  ˆj  ˆi  kˆ  ˆj  kˆ  1 1  cos 90   0
5
© 2007 S. Kenny, Ph.D., P.Eng.
Go to zero
ENGI 1313 Statics I – Lecture 07

Application of Dot Product

Angle between two vectors

Cables forces and the pole?
•
• and ?
 
 
A  B  A B cos 

1 
  cos 


 
if A  B  0
6
 
A B
 
A B
0    180



 cos



then   cos
© 2007 S. Kenny, Ph.D., P.Eng.
1
1





Component magnitudes
Ax B x  Ay B y  AzB z
Ax  Ay  Az
2
0   90

2

2

Bx  By  Bz
2
AB
ENGI 1313 Statics I – Lecture 07
2
2




Application of Dot Product (cont.)

Component magnitude
of A on a parallel or
collinear line
with line aa
If A|| has + sense then
same direction as ^u

 
A ||  A cos   A  u

Recall
 
 
A  B  A B cos 
7
Component A||

A ||  A cos 
 

B  uˆ and B  1
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07
Application of Dot Product (cont.)

The vector A|| can
be determined by:

A || 



  
 
A cos  u u  A  u u

Application of Dot Product
for Component A||
8
© 2007 S. Kenny, Ph.D., P.Eng.

Vector A||
Multiply by Unit Vector û
to obtain Vector A||
ENGI 1313 Statics I – Lecture 07
Application of Dot Product (cont.)

For force vector F at
Point A: What is the
component magnitude
parallel (|F1|) to the
pipe (OA)?


A ||  A cos   A  uˆ




F1  F|| OA  F cos   F  uˆ
9
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07
Application of Dot Product (cont.)

For force vector F at
Point A: what is the
component magnitude
perpendicular (F2) to the
pipe (OA)?

Method 1
 
ˆ
1  F  u
  cos  
 F




F  F2  F sin 





F1  F|| OA  F1 cos   F  uˆ
Method 2


F   F2 
10






F
2

 F ||
2

© 2007 S. Kenny, Ph.D., P.Eng.

F
2

 F1
2
ENGI 1313 Statics I – Lecture 07
Comprehension Quiz 7-01

The dot product of two vectors results in a
 
 
_________ quantity.
A  B  A B cos 
A) scalar
 B) vector
 C) complex number
 D) unit vector


11
Answer: A
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07
Example Problem 7-01

12
For the Cartesian force
vector, find the angle
between the force
vector and the pole,
and the magnitude of
the projection of the
force along the pole OA
© 2007 S. Kenny, Ph.D., P.Eng.
A
ENGI 1313 Statics I – Lecture 07
Example Problem 7-01 (cont.) 

 
A  B  A B cos 

Position vector rOA



rOA  2 ˆi  2 ˆj  1 kˆ m

Magnitude of |rOA|

rOA 
2
2
 2
2
 1 m  3m
2
A

Magnitude of |F|

F 
13
2
2
4
2
 10
2
 10 . 95 kN
© 2007 S. Kenny, Ph.D., P.Eng.

1 
  cos 


ENGI 1313 Statics I – Lecture 07
 
F  rOA
 
F rOA





Example Problem 7-01 (cont.) 

 
A  B  A B cos 

Find the angle 
between rOA and F

1 
  cos 


 
A B
 
A B






Ax  Ay  Az
2
2
2
Bx  By  Bz
2
2

F x rOA x  F y rOAy  F z rOAz
1 
  cos 
 

F rOA

  cos
14
1

Ax B x  Ay B y  AzB z
2





  2 2   4 2   10   1   kN  m

10 . 95 kN 3 m 

© 2007 S. Kenny, Ph.D., P.Eng.
A

  86 . 5 

ENGI 1313 Statics I – Lecture 07
Example Problem 7-01 (cont.)

Find magnitude of
the projection of the
force F along the
pole OA




F|| OA  F cos   F  uˆ


F|| OA  10 . 95 kN cos 86 . 51  0 . 667 kN

rOA
2 ˆi  2 ˆj  1 kˆ m
uˆ OA  

rOA
3m

A


2 2   4 2   10   1 
F|| OA 
kN  0 . 667 kN
3
15
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07
Comprehension Quiz 7-02

If the dot product of two non-zero vectors
is 0, then the two vectors must be ______
to each other.
A) parallel (pointing in the same direction)
 B) parallel (pointing in the opposite direction)
 C) perpendicular
 
 
A  B  A B cos 
 D) cannot be determined.


16
Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07
Comprehension Quiz 7-03

The Dot product can be used
to
find
all
of
 
 
the following except ____ A  B  A B cos 
A) sum of two vectors
 B) angle between two vectors
 C) vector component parallel to a line
 D) vector component perpendicular to a line


17
Answer: A
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07
Comprehension Quiz 7-04

Find the dot product (PQ) for

P  5 ˆi  2 ˆj  3 kˆ m







18

 
 
A  B  A B cos 

Q   2 ˆi  5 ˆj  4 kˆ m

A) -12 m
B) 12 m
C) 12 m2
D) -12 m2
E) 10 m2
Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07

Classification of Textbook Problems
Hibbeler (2007)
19
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07
References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1

20
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 07