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ENGI 1313 Mechanics I . Lecture 07: Vector Dot Product. 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

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engi 1313 mechanics i

ENGI 1313 Mechanics I

Lecture 07: Vector Dot Product

chapter 2 objectives
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
lecture 07 objectives
Lecture 07 Objectives
  • to examine the concept of dot product
overview of dot product
Overview of Dot Product
  • Definition
  • Laws of Operations
    • Commutative law
    • Scalar Multiplication
    • Distributive law
overview of dot product cont
Overview of Dot Product (cont.)
  • Dot Product of Cartesian Vectors

Go to zero

application of dot product
Application of Dot Product
  • Angle between two vectors
    • Cables forces and the pole?
      • and ?

Component magnitudes

application of dot product cont
If A||has + sense then same direction as u

^

Application of Dot Product (cont.)
  • Component magnitudeof A on a parallel or collinear linewith line aa
    • Recall

Component A||

application of dot product cont1
Application of Dot Product (cont.)
  • The vector A|| canbe determined by:

Vector A||

Application of Dot Product for Component A||

Multiply by Unit Vector ûto obtain Vector A||

application of dot product cont2
Application of Dot Product (cont.)
  • For force vector F at Point A: What is the component magnitudeparallel (|F1|) to the pipe (OA)?
application of dot product cont3
Application of Dot Product (cont.)
  • For force vector F at Point A: what is the component magnitudeperpendicular (F2) to the pipe (OA)?
    • Method 1
    • Method 2
comprehension quiz 7 01
Comprehension Quiz 7-01
  • The dot product of two vectors results in a _________ quantity.
    • A) scalar
    • B) vector
    • C) complex number
    • D) unit vector
  • Answer: A
example problem 7 01
AExample Problem 7-01
  • 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
example problem 7 01 cont
AExample Problem 7-01 (cont.)
  • Position vector rOA
  • Magnitude of |rOA|
  • Magnitude of |F|
example problem 7 01 cont1
AExample Problem 7-01 (cont.)
  • Find the angle between rOA and F

example problem 7 01 cont2
AExample Problem 7-01 (cont.)
  • Find magnitude of the projection of the force F along the pole OA

comprehension quiz 7 02
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
    • D) cannot be determined.
  • Answer: C
comprehension quiz 7 03
Comprehension Quiz 7-03
  • The Dot product can be used to find all of the following except ____
    • A) sum of two vectors
    • B) angle between two vectors
    • C) vector component parallel to a line
    • D) vector component perpendicular to a line
  • Answer: A
comprehension quiz 7 04
Comprehension Quiz 7-04
  • Find the dot product (PQ) for
    • A) -12 m
    • B) 12 m
    • C) 12 m2
    • D) -12 m2
    • E) 10 m2
  • Answer: C
references
References
  • Hibbeler (2007)
  • http://wps.prenhall.com/esm_hibbeler_engmech_1
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