# ENGI 1313 Mechanics I - PowerPoint PPT Presentation

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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

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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

• to examine the concept of dot product

### Lecture 07 Objectives

• to examine the concept of dot product

### Overview of Dot Product

• Definition

• Laws of Operations

• Commutative law

• Scalar Multiplication

• Distributive law

### Overview of Dot Product (cont.)

• Dot Product of Cartesian Vectors

Go to zero

### Application of Dot Product

• Angle between two vectors

• Cables forces and the pole?

• and ?

Component magnitudes

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 (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 (cont.)

• For force vector F at Point A: What is the component magnitudeparallel (|F1|) to the pipe (OA)?

### 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

• The dot product of two vectors results in a _________ quantity.

• A) scalar

• B) vector

• C) complex number

• D) unit vector

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

A

### Example Problem 7-01 (cont.)

• Position vector rOA

• Magnitude of |rOA|

• Magnitude of |F|

A

### Example Problem 7-01 (cont.)

• Find the angle between rOA and F

A

### Example Problem 7-01 (cont.)

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

### 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.

### 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

### 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
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