ENGI 1313 Mechanics I

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ENGI 1313 Mechanics I

• Lecture 07 Vector Dot Product

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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

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Lecture 07 Objectives

• to examine the concept of dot product

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Overview of Dot Product

• Definition
• Laws of Operations
• Commutative law
• Scalar Multiplication
• Distributive law

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Overview of Dot Product (cont.)

• Dot Product of Cartesian Vectors

Go to zero
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Application of Dot Product

• Angle between two vectors
• Cables forces and the pole?
• ?
• and ??

Component magnitudes
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Application of Dot Product (cont.)

• Component magnitudeof A on a parallel or
  collinear linewith line aa?
• Recall

Component A
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Application of Dot Product (cont.)

• The vector A canbe determined by

Vector A
Application of Dot Product for Component A
Multiply by Unit Vector û to obtain Vector A
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Application of Dot Product (cont.)

• For force vector F at Point A What is the
  component magnitude parallel (F1) to the
  pipe (OA)?

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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
• Method 2

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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

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Example 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

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Example Problem 7-01 (cont.)

• Position vector rOA
• Magnitude of rOA
• Magnitude of F

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Example Problem 7-01 (cont.)

• Find the angle ? between rOA and F

?
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Example Problem 7-01 (cont.)

• Find magnitude of the projection of the force F
  along the pole OA

?
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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

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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

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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

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Classification of Textbook Problems
Hibbeler (2007)
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References

• Hibbeler (2007)
• http//wps.prenhall.com/esm_hibbeler_engmech_1