1. Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca ENGI 1313 Mechanics I Lecture 07:Vector Dot Product

2. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 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. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 3 Lecture 07 Objectives to examine the concept of dot product

4. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 4 Overview of Dot Product Definition Laws of Operations  Commutative law  Scalar Multiplication  Distributive law

5. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 5 Overview of Dot Product (cont.) Dot Product of Cartesian Vectors Go to zero

6. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 6 Application of Dot Product Angle between two vectors  Cables forces and the pole?  and  ? Component magnitudes

7. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 7 Application of Dot Product (cont.) Component magnitude of A on a parallel or collinear line with line aa  Recall Component A || If A|| has + sense then same direction as u ^

8. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 8 Application of Dot Product (cont.) The vector A || can be determined by: Vector A || Application of Dot Product for Component A || Multiply by Unit Vector û to obtain Vector A ||

9. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 9 Application of Dot Product (cont.) For force vector F at Point A: What is the component magnitude parallel (|F 1 |) to the pipe (OA)?

10. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 10 Application of Dot Product (cont.) For force vector F at Point A: what is the component magnitude perpendicular (F 2 ) to the pipe (OA)?  Method 1  Method 2

11. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 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. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 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. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 13 A Example Problem 7-01 (cont.) Position vector r OA Magnitude of |r OA | Magnitude of |F|

14. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 14 A Example Problem 7-01 (cont.) Find the angle  between r OA and F 

15. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 15 A Example Problem 7-01 (cont.) Find magnitude of the projection of the force F along the pole OA 

16. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 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. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 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. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 18 Comprehension Quiz 7-04 Find the dot product (P  Q) for  A) -12 m  B) 12 m  C) 12 m 2  D) -12 m 2  E) 10 m 2 Answer: C

19. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 19 Classification of Textbook Problems Hibbeler (2007)

20. ENGI 1313 Statics I – Lecture 07© 2007 S. Kenny, Ph.D., P.Eng. 20 References Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_eng mech_1