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ENGI 1313 Mechanics I

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

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

  2. 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

  3. Lecture 07 Objectives • to examine the concept of dot product

  4. Overview of Dot Product • Definition • Laws of Operations • Commutative law • Scalar Multiplication • Distributive law

  5. Overview of Dot Product (cont.) • Dot Product of Cartesian Vectors Go to zero

  6. Application of Dot Product • Angle between two vectors • Cables forces and the pole? •  • and ? Component magnitudes

  7. 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||

  8. 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||

  9. Application of Dot Product (cont.) • For force vector F at Point A: What is the component magnitudeparallel (|F1|) to the pipe (OA)?

  10. 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

  11. 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

  12. A 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

  13. A Example Problem 7-01 (cont.) • Position vector rOA • Magnitude of |rOA| • Magnitude of |F|

  14. A Example Problem 7-01 (cont.) • Find the angle between rOA and F 

  15. A Example Problem 7-01 (cont.) • Find magnitude of the projection of the force F along the pole OA 

  16. 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

  17. 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

  18. 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

  19. Classification of Textbook Problems Hibbeler (2007)

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

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