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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

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


Example problem 7 01 cont

A

Example Problem 7-01 (cont.)

  • Position vector rOA

  • Magnitude of |rOA|

  • Magnitude of |F|


Example problem 7 01 cont1

A

Example Problem 7-01 (cont.)

  • Find the angle between rOA and F


Example problem 7 01 cont2

A

Example 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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