( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) ( )

Created by T. Madas VECTOR MOMENTS. Created by T. Madas

VECTOR MOMENTS Question 1 (**) The vectors i, j and k are unit vectors mutually perpendicular to one another. Relative to a fixed origin O, a light rigid rod has its ends located at the points 0, 7,4 B

TABLE OF CONTENTS 2 CHAPTER 1

TABLE OF CONTENTS CHAPTER 1 Quadratics CHAPTER Functions 3 CHAPTER 3 Coordinate Geometry 3 CHAPTER 4 Circular Measure 4 CHAPTER 5 Trigonometry 4 CHAPTER 6 Vectors 5 CHAPTER 7 Series 6 CHAPTER 8 Differentiation

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2

Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1

Vectors in Physics. Topics to review:

Vectors in Physics Topics to review: Scalars Versus Vectors The Components of a Vector Adding and Subtracting Vectors Unit Vectors Position, Displacement, Velocity, and Acceleration Vectors Relative Motion

Dot Product August 2013

Dot Product 12.3 30 August 2013 Dot product. v = v 1, v 2,..., v n, w = w 1, w 2,..., w n The dot product v w is v w = v 1 w 1 + v 2 w 2 + + v n w n n = v i w i. i=1 Example: 1, 4, 5 2, 8, 0 = 1 2 + 4

Find the component form of with initial point A(1, 3) and terminal point B(1, 3). Component form = 1 1, 3 ( 3) (x 1., y 1. ) = (1, 3) = 0, 6 Subtract.

Express a Vector in Component Form Find the component form of with initial point A(1, 3) and terminal point B(1, 3). = x 2 x 1, y 2 y 1 Component form = 1 1, 3 ( 3) (x 1, y 1 ) = (1, 3) and ( x 2, y 2

EXAM 1. OPEN BOOK AND CLOSED NOTES Thursday, February 18th, 2010

ME 35 - Machine Design I Spring Semester 010 Name of Student Lab. Div. Number EXAM 1. OPEN BOOK AND CLOSED NOTES Thursday, February 18th, 010 Please use the blank paper provided for your solutions. Write

When two letters name a vector, the first indicates the and the second indicates the of the vector.

8-8 Chapter 8 Applications of Trigonometry 8.3 Vectors, Operations, and the Dot Product Basic Terminology Algeraic Interpretation of Vectors Operations with Vectors Dot Product and the Angle etween Vectors

Vectors. J.R. Wilson. September 27, 2018

Vectors J.R. Wilson September 27, 2018 This chapter introduces vectors that are used in many areas of physics (needed for classical physics this year). One complication is that a number of different forms

Vectors. J.R. Wilson. September 28, 2017

Vectors J.R. Wilson September 28, 2017 This chapter introduces vectors that are used in many areas of physics (needed for classical physics this year). One complication is that a number of different forms

Chapter 2 One-Dimensional Kinematics

Review: Chapter 2 One-Dimensional Kinematics Description of motion in one dimension Copyright 2010 Pearson Education, Inc. Review: Motion with Constant Acceleration Free fall: constant acceleration g =

Section 8.2 Vector Angles

Section 8.2 Vector Angles INTRODUCTION Recall that a vector has these two properties: 1. It has a certain length, called magnitude 2. It has a direction, indicated by an arrow at one end. In this section

7.1 Projections and Components

7. Projections and Components As we have seen, the dot product of two vectors tells us the cosine of the angle between them. So far, we have only used this to find the angle between two vectors, but cosines

Math 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.

Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =

VECTORS IN A STRAIGHT LINE

A. The Equation of a Straight Line VECTORS P3 VECTORS IN A STRAIGHT LINE A particular line is uniquely located in space if : I. It has a known direction, d, and passed through a known fixed point, or II.

9.5. Lines and Planes. Introduction. Prerequisites. Learning Outcomes

Lines and Planes 9.5 Introduction Vectors are very convenient tools for analysing lines and planes in three dimensions. In this Section you will learn about direction ratios and direction cosines and then

Vectors. AP/Honors Physics Mr. Velazquez

Vectors AP/Honors Physics Mr. Velazquez The Basics Any quantity that refers to a magnitude and a direction is known as a vector quantity. Velocity, acceleration, force, momentum, displacement Other quantities

Equivalent Force Systems

Equivalent Force Systems EQUIVALENT SYSTEMS for SINGLE FORCE Determining the effect of moving a force. 1. MOVING A FORCE ON ITS LINE OF ACTION 2. MOVING A FORCE OFF OF ITS LINE OF ACTION Equivalent Force

Chapter 6 Additional Topics in Trigonometry, Part II

Chapter 6 Additional Topics in Trigonometry, Part II Section 3 Section 4 Section 5 Vectors in the Plane Vectors and Dot Products Trigonometric Form of a Complex Number Vocabulary Directed line segment

Topic Learning Outcomes Suggested Teaching Activities Resources On-Line Resources

UNIT 3 Trigonometry and Vectors (P1) Recommended Prior Knowledge. Students will need an understanding and proficiency in the algebraic techniques from either O Level Mathematics or IGCSE Mathematics. Context.

CALCULUS AB/BC SUMMER REVIEW PACKET (Answers)

Name CALCULUS AB/BC SUMMER REVIEW PACKET (Answers) I. Simplify. Identify the zeros, vertical asymptotes, horizontal asymptotes, holes and sketch each rational function. Show the work that leads to your

( ) 2. = kq 1 q 2 r 2. Analysis: F E. k q 1. Solution: F E % 8.99!10 9 ( (1.00!10 )4 C )(1.00!10 )5 C ) $ (2.00 m) 2 F E

Section 7.: Coulomb s Law Tutorial 1 Practice, page 33 1. Given: q 1 1.00 10 4 C; q 1.00 10 5 C; r.00 m; k 8.99 10 9 N m /C Required: Analysis: Solution: # N m % 8.99 10 9 ( (1.00 10 )4 C )(1.00 10 )5

UNIT 1 VECTORS INTRODUCTION 1.1 OBJECTIVES. Stucture

UNIT 1 VECTORS 1 Stucture 1.0 Introduction 1.1 Objectives 1.2 Vectors and Scalars 1.3 Components of a Vector 1.4 Section Formula 1.5 nswers to Check Your Progress 1.6 Summary 1.0 INTRODUCTION In this unit,

F A C U L T Y O F E D U C A T I O N. Physics Electromagnetism: Induced Currents Science and Mathematics Education Research Group

F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagogy Physics Electromagnetism: Induced Currents Science and Mathematics Education Research Group Supported by UBC Teaching and Learning

Chapter 4. Motion in Two Dimensions. Professor Wa el Salah

Chapter 4 Motion in Two Dimensions Kinematics in Two Dimensions Will study the vector nature of position, velocity and acceleration in greater detail. Will treat projectile motion and uniform circular

6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line

CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle

Mathematics 2203, Test 1 - Solutions

Mathematics 220, Test 1 - Solutions F, 2010 Philippe B. Laval Name 1. Determine if each statement below is True or False. If it is true, explain why (cite theorem, rule, property). If it is false, explain

Module 10 Polar Form of Complex Numbers

MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex

VECTORS. Vectors OPTIONAL - I Vectors and three dimensional Geometry

Vectors OPTIONAL - I 32 VECTORS In day to day life situations, we deal with physical quantities such as distance, speed, temperature, volume etc. These quantities are sufficient to describe change of position,

CHAPTER 10 VECTORS POINTS TO REMEMBER

For more important questions visit : www4onocom CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b

vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component

MR. YATES. Vocabulary. Quadratic Cubic Monomial Binomial Trinomial Term Leading Term Leading Coefficient

ALGEBRA II WITH TRIGONOMETRY COURSE OUTLINE SPRING 2009. MR. YATES Vocabulary Unit 1: Polynomials Scientific Notation Exponent Base Polynomial Degree (of a polynomial) Constant Linear Quadratic Cubic Monomial

BSc (Hons) in Computer Games Development. vi Calculate the components a, b and c of a non-zero vector that is orthogonal to

1 APPLIED MATHEMATICS INSTRUCTIONS Full marks will be awarded for the correct solutions to ANY FIVE QUESTIONS. This paper will be marked out of a TOTAL MAXIMUM MARK OF 100. Credit will be given for clearly

Vectors and Matrices Lecture 2

Vectors and Matrices Lecture 2 Dr Mark Kambites School of Mathematics 13/03/2014 Dr Mark Kambites (School of Mathematics) COMP11120 13/03/2014 1 / 20 How do we recover the magnitude of a vector from its

Solutions for Chapter 3

Solutions for Chapter Solutions for exercises in section 0 a X b x, y 6, and z 0 a Neither b Sew symmetric c Symmetric d Neither The zero matrix trivially satisfies all conditions, and it is the only possible

Vectors for Physics. AP Physics C

Vectors for Physics AP Physics C A Vector is a quantity that has a magnitude (size) AND a direction. can be in one-dimension, two-dimensions, or even three-dimensions can be represented using a magnitude

CSC 470 Introduction to Computer Graphics. Mathematical Foundations Part 2

CSC 47 Introduction to Computer Graphics Mathematical Foundations Part 2 Vector Magnitude and Unit Vectors The magnitude (length, size) of n-vector w is written w 2 2 2 w = w + w2 + + w n Example: the

Math 215 HW #9 Solutions

Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith

Vectors Summary. can slide along the line of action. not restricted, defined by magnitude & direction but can be anywhere.

Vectors Summary A vector includes magnitude (size) and direction. Academic Skills Advice Types of vectors: Line vector: Free vector: Position vector: Unit vector (n ): can slide along the line of action.

MT225 Homework 6 - Solution. Problem 1: [34] 5 ft 5 ft 5 ft A B C D. 12 kips. 10ft E F G

MT225 Homework 6 - Solution Problem 1: [34] 5 ft 5 ft 5 ft C D 10ft 12 kips E G To begin, we need to find the reaction forces at supports and E. To do so, we draw a free-body diagram of the entire structure

12.3 Dot Products, 12.4 Cross Products

12.3 Dot Products, 12.4 Cross Products How do we multiply vectors? How to multiply vectors is not at all obvious, and in fact, there are two different ways to make sense of vector multiplication, each

RELATIVE MOTION ANALYSIS (Section 12.10)

RELATIVE MOTION ANALYSIS (Section 1.10) Today s Objectives: Students will be able to: a) Understand translating frames of reference. b) Use translating frames of reference to analyze relative motion. APPLICATIONS

VECTORS. Section 6.3 Precalculus PreAP/Dual, Revised /11/ :41 PM 6.3: Vectors in the Plane 1

VECTORS Section 6.3 Precalculus PreAP/Dual, Revised 2017 Viet.dang@humbleisd.net 10/11/2018 11:41 PM 6.3: Vectors in the Plane 1 DEFINITIONS A. Vector is used to indicate a quantity that has both magnitude

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 3 - Thurs 5th Oct 2017 Vectors and 3D geometry

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 3 - Thurs 5th Oct 2017 Vectors and 3D geometry So far, all our calculus has been two-dimensional, involving only x and y. Nature is threedimensional,

Written as per the revised G Scheme syllabus prescribed by the Maharashtra State Board of Technical Education (MSBTE) w.e.f. academic year

Written as per the revised G Scheme syllabus prescribed by the Maharashtra State Board of Technical Education (MSBTE) w.e.f. academic year 2012-2013 Basic MATHEMATICS First Year Diploma Semester - I First

Calculus II - Basic Matrix Operations

Calculus II - Basic Matrix Operations Ryan C Daileda Terminology A matrix is a rectangular array of numbers, for example 7,, 7 7 9, or / / /4 / / /4 / / /4 / /6 The numbers in any matrix are called its

APPLIED MECHANICS I Resultant of Concurrent Forces Consider a body acted upon by co-planar forces as shown in Fig 1.1(a).

PPLIED MECHNICS I 1. Introduction to Mechanics Mechanics is a science that describes and predicts the conditions of rest or motion of bodies under the action of forces. It is divided into three parts 1.

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b

vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component

The Cross Product of Two Vectors

The Cross roduct of Two Vectors In proving some statements involving surface integrals, there will be a need to approximate areas of segments of the surface by areas of parallelograms. Therefore it is

Vectors. However, cartesian coordinates are really nothing more than a way to pinpoint an object s position in space

Vectors Definition of Scalars and Vectors - A quantity that requires both magnitude and direction for a complete description is called a vector quantity ex) force, velocity, displacement, position vector,

The Cross Product. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) The Cross Product Spring /

The Cross Product Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) The Cross Product Spring 2012 1 / 15 Introduction The cross product is the second multiplication operation between vectors we will

MULTIPLE PRODUCTS OBJECTIVES. If a i j,b j k,c i k, = + = + = + then a. ( b c) ) 8 ) 6 3) 4 5). If a = 3i j+ k and b 3i j k = = +, then a. ( a b) = ) 0 ) 3) 3 4) not defined { } 3. The scalar a. ( b c)

MathCity.org Merging man and maths

Mathity.org Merging man and maths Exercise 7.3 (s)page 349 alculus and nalytic Geometry, MTHEMTIS 1 @ http://www.mathcity.org, Version vailable online: 3.0 Question # 1 Find the cosine of the angle θ between

2.1 Scalars and Vectors

2.1 Scalars and Vectors Scalar A quantity characterized by a positive or negative number Indicated by letters in italic such as A e.g. Mass, volume and length 2.1 Scalars and Vectors Vector A quantity

Chapter 4. Motion in Two Dimensions

Chapter 4 Motion in Two Dimensions Kinematics in Two Dimensions Will study the vector nature of position, velocity and acceleration in greater detail Will treat projectile motion and uniform circular motion

SUPPLEMENT I. Example. Graph the vector 4, 3. Definition. Given two points A(x 1, y 1 ) and B(x 2, y 2 ), the vector represented by # AB is # AB =,

SUPPLEMENT I 1. Vectors Definition. A vector is a quantity that has both a magnitude and a direction. A twodimensional vector is an ordered pair a = a 1, a 2 of real numbers. The numbers a 1 and a 2 are

HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)

HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe

F F. proj cos( ) v. v proj v

Geometric Definition of Dot Product 1.2 The Dot Product Suppose you are pulling up on a rope attached to a box, as shown above. How would you find the force moving the box towards you? As stated above,

xvi xxiii xxvi Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7

About the Author v Preface to the Instructor xvi WileyPLUS xxii Acknowledgments xxiii Preface to the Student xxvi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real

Exercise Solutions for Introduction to 3D Game Programming with DirectX 10

Exercise Solutions for Introduction to 3D Game Programming with DirectX 10 Frank Luna, September 6, 009 Solutions to Part I Chapter 1 1. Let u = 1, and v = 3, 4. Perform the following computations and

Summer Mathematics Prep

Summer Mathematics Prep Entering Calculus Chesterfield County Public Schools Department of Mathematics SOLUTIONS Domain and Range Domain: All Real Numbers Range: {y: y } Domain: { : } Range:{ y : y 0}

Phys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole

Phs 221 Chapter 3 Vectors adzubenko@csub.edu http://www.csub.edu/~adzubenko 2014. Dzubenko 2014 rooks/cole 1 Coordinate Sstems Used to describe the position of a point in space Coordinate sstem consists

MATH 109 TOPIC 3 RIGHT TRIANGLE TRIGONOMETRY. 3a. Right Triangle Definitions of the Trigonometric Functions

Math 09 Ta-Right Triangle Trigonometry Review Page MTH 09 TOPIC RIGHT TRINGLE TRIGONOMETRY a. Right Triangle Definitions of the Trigonometric Functions a. Practice Problems b. 5 5 90 and 0 60 90 Triangles

OpenStax-CNX module: m Vectors. OpenStax College. Abstract

OpenStax-CNX module: m49412 1 Vectors OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section you will: Abstract View vectors

Math 259 Winter Solutions to Homework # We will substitute for x and y in the linear equation and then solve for r. x + y = 9.

Math 59 Winter 9 Solutions to Homework Problems from Pages 5-5 (Section 9.) 18. We will substitute for x and y in the linear equation and then solve for r. x + y = 9 r cos(θ) + r sin(θ) = 9 r (cos(θ) +

Chapter 7 Introduction to vectors

Introduction to ectors MC Qld-7 Chapter 7 Introduction to ectors Eercise 7A Vectors and scalars a i r + s ii r s iii s r b i r + s Same as a i ecept scaled by a factor of. ii r s Same as a ii ecept scaled

The Cross Product. In this section, we will learn about: Cross products of vectors and their applications.

The Cross Product In this section, we will learn about: Cross products of vectors and their applications. THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is a

27 ft 3 adequately describes the volume of a cube with side 3. ft F adequately describes the temperature of a person.

VECTORS The stud of ectors is closel related to the stud of such phsical properties as force, motion, elocit, and other related topics. Vectors allow us to model certain characteristics of these phenomena

Section 13.4 The Cross Product

Section 13.4 The Cross Product Multiplying Vectors 2 In this section we consider the more technical multiplication which can be defined on vectors in 3-space (but not vectors in 2-space). 1. Basic Definitions

9709 MATHEMATICS. 9709/32 Paper 3 (Pure Mathematics), maximum raw mark 75

CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International Advanced Subsidiary and Advanced Level MARK SCHEME for the March 06 series 9709 MATHEMATICS 9709/3 Paper 3 (Pure Mathematics), maximum raw mark

Introduction to Vector Functions

Introduction to Vector Functions Limits and Continuity Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Introduction to Vector Functions Spring 2012 1 / 14 Introduction In this section, we study

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: 2 x 3 + 3

Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: x 3 + 3 x + x + 3x 7 () x 3 3x + x 3 From the standpoint of integration, the left side of Equation

CALCULUS AB/BC SUMMER REVIEW PACKET

Name CALCULUS AB/BC SUMMER REVIEW PACKET Welcome to AP Calculus! Calculus is a branch of advanced mathematics that deals with problems that cannot be solved with ordinary algebra such as rate problems

ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICS GCE Ordinary Level (Syllabus 4018) CONTENTS Page NOTES 1 GCE ORDINARY LEVEL ADDITIONAL MATHEMATICS 4018 2 MATHEMATICAL NOTATION 7 4018 ADDITIONAL MATHEMATICS O LEVEL (2009) NOTES

EGR 1301 Introduction to Static Analysis

Slide 1 EGR 1301 Introduction to Static Analysis Presentation adapted from Distance Learning / Online Instructional Presentation Originally created by Mr. Dick Campbell Presented by: Departments of Engineering

EXAM 1. OPEN BOOK AND CLOSED NOTES.

ME 35 - Machine Design I Summer Semester 013 Name of Student Lab Section Number EXAM 1. OPEN BOOK AND CLOSED NOTES. Wednesday, June 6th, 013 Use the blank paper provided for your solutions. Write on one

Unit #17: Spring Trig Unit. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that same amount.

Name Unit #17: Spring Trig Unit Notes #1: Basic Trig Review I. Unit Circle A circle with center point and radius. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that

GCE214 Applied Mechanics-Statics. Lecture 04: 27/09/2017

GCE214 Applied Mechanics-Statics Lecture 04: 27/09/2017 Dr. Ayokunle O. Balogun balogun.ayokunle@lmu.edu.ng Class: Wednesday (3 5 pm) Venue: LT1 Etiquettes and MOP Attendance is a requirement. There may

SECTION 6.3: VECTORS IN THE PLANE

(Section 6.3: Vectors in the Plane) 6.18 SECTION 6.3: VECTORS IN THE PLANE Assume a, b, c, and d are real numbers. PART A: INTRO A scalar has magnitude but not direction. We think of real numbers as scalars,

Trigonometry I. Pythagorean theorem: WEST VIRGINIA UNIVERSITY Physics

Trigonometry I Pythagorean theorem: Trigonometry II 90 180 270 360 450 540 630 720 sin(x) and cos(x) are mathematical functions that describe oscillations. This will be important later, when we talk about

MATHEMATICS (SET -3) Labour cost Z 300x 400y (to be minimized) The constraints are: SECTION - A 1. f (x) is continuous at x 3 f (3) lim f (x)

8 Class th (SET -) BD PPER -7 M T H E M T I C S () SECTION -. f () is continuous at f () lim f () ( ) 6 k lim ( )( 6) k lim ( ) k. adj I 8 I 8 I 8I 8. P : z 5 5 P : 5 5z z 8 Distance between P & P sin

Rigid Body Dynamics, SG2150 Solutions to Exam,

KTH Mechanics 011 10 Calculational problems Rigid Body Dynamics, SG150 Solutions to Eam, 011 10 Problem 1: A slender homogeneous rod of mass m and length a can rotate in a vertical plane about a fied smooth

or i 2 = -1 i 4 = 1 Example : ( i ),, 7 i and 0 are complex numbers. and Imaginary part of z = b or img z = b

1 A- LEVEL MATHEMATICS P 3 Complex Numbers (NOTES) 1. Given a quadratic equation : x 2 + 1 = 0 or ( x 2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose

Physics Dynamics: Forces. Science and Mathematics Education Research Group

F F CULTY C U L T Y OF O F EDUCTION E D U C T I O N Department of Curriculum and Pedagogy Physics Dynamics: Forces Science and Mathematics Education Research Group Supported by UBC Teaching and Learning

Chapter 2 A Mathematical Toolbox

Chapter 2 Mathematical Toolbox Vectors and Scalars 1) Scalars have only a magnitude (numerical value) Denoted by a symbol, a 2) Vectors have a magnitude and direction Denoted by a bold symbol (), or symbol

VECTORS IN THREE DIMENSIONS

1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude

Omm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics

LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector

Analytic Trigonometry. Copyright Cengage Learning. All rights reserved.

Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.4 Basic Trigonometric Equations Copyright Cengage Learning. All rights reserved. Objectives Basic Trigonometric Equations Solving

TRIGONOMETRY INTRODUCTION. Objectives. SESSION 1-5 ANGLES A positive angle measures a rotation in an anticlockwise direction.

TRIGONOMETRY INTRODUCTION s the title of the unit suggests, it deals with the calculation of angles or the length of their sides. In this unit, the trigonometric ratios of acute angles, general angles

Vectors. Examples of vectors include: displacement, velocity, acceleration, and force. Examples of scalars include: distance, speed, time, and volume.

Math 150 Prof. Beydler 7.4/7.5 Notes Page 1 of 6 Vectors Suppose a car is heading NE (northeast) at 60 mph. We can use a vector to help draw a picture (see right). v A vector consists of two parts: 1.

e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)

Chapter 0 Application of differential calculus 014 GDC required 1. Consider the curve with equation f () = e for 0. Find the coordinates of the point of infleion and justify that it is a point of infleion.

Lecture Notes 3

12.005 Lecture Notes 3 Tensors Most physical quantities that are important in continuum mechanics like temperature, force, and stress can be represented by a tensor. Temperature can be specified by stating

Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.

Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read

Congruence Axioms. Data Required for Solving Oblique Triangles

Math 335 Trigonometry Sec 7.1: Oblique Triangles and the Law of Sines In section 2.4, we solved right triangles. We now extend the concept to all triangles. Congruence Axioms Side-Angle-Side SAS Angle-Side-Angle

7.2 Matrix Algebra. DEFINITION Matrix. D 21 a 22 Á a 2n. EXAMPLE 1 Determining the Order of a Matrix d. (b) The matrix D T has order 4 * 2.

530 CHAPTER 7 Systems and Matrices 7.2 Matrix Algebra What you ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix

Engineering Mechanics: Statics in SI Units, 12e Force Vectors

Engineering Mechanics: Statics in SI Units, 1e orce Vectors 1 Chapter Objectives Parallelogram Law Cartesian vector form Dot product and angle between vectors Chapter Outline 1. Scalars and Vectors. Vector

STEP II, ax y z = 3, 2ax y 3z = 7, 3ax y 5z = b, (i) In the case a = 0, show that the equations have a solution if and only if b = 11.

STEP II, 2003 2 Section A: Pure Mathematics 1 Consider the equations ax y z = 3, 2ax y 3z = 7, 3ax y 5z = b, where a and b are given constants. (i) In the case a = 0, show that the equations have a solution

Physics 40 HW#3 Chapter 3 Problems from Knight you should do (but don t turn in): Ch 3 Problems: 7, 10, 15, 23, 33, 37, 40, 42, 43

Physics 40 HW#3 Chapter 3 Problems from Knight you should do (but don t turn in): Ch 3 Problems: 7, 10, 15, 23, 33, 37, 40, 42, 43 7, 10, 15, 23, 33, 37, 40, 42, 43 3.7. Visualiz e: Solve: (a) v = (10

Calculus with Analytic Geometry I Exam 10, Take Home Friday, November 8, 2013 Solutions.

All exercises are from Section 4.7 of the textbook. 1. Calculus with Analytic Geometry I Exam 10, Take Home Friday, November 8, 2013 Solutions. 2. Solution. The picture suggests using the angle θ as variable;

General Physics I, Spring Vectors

General Physics I, Spring 2011 Vectors 1 Vectors: Introduction A vector quantity in physics is one that has a magnitude (absolute value) and a direction. We have seen three already: displacement, velocity,