# Section V.3: Dot Product

Size: px
Start display at page:

Transcription

1 Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here, since Physics 191 has a unit on vector addition/subtraction. We will look at the dot product, which is also known as the inner product or scalar product. Definition The dot product of vectors A and B is written as A B. For two-dimensional vectors, the dot product is calculated by multiplying the components of A and B together as follows: A B = A x B x + A y B y and since A x, A y, B x, and B y are all just real numbers, the dot product is also a real number and is consequently a scalar. (This is why it s also called the scalar product.) A has components A x = 12, A y = 5. B has components B x = 2, B y = 6. A B = A x B x + A y B y = ( 6) = = 6. Calculate the dot product B A of the vectors in the preceding example. B A = B x A x + B y A y = ( 6) 5 = = 6.

2 Page 2 Math 185 Vectors A = 3i 5j B = 2i 7j A B = A x B x + A y B y = 3 ( 2) + ( 5) ( 7) = = 29. You can see from these examples that the result of the dot product, a scalar, can be positive or negative. You can also see that in the first two examples, A B = B A. Does this work for all vectors A and B? Let s look at the calculation: A x B x + A y B y. Since multiplication of real numbers is commutative (order doesn t matter), then A x B x + A y B y = B x A x + B y A y for any vectors A and B. Therefore, A B = B A for all vectors A and B. Properties of the Dot Product We ve just established that A B = B A. What are some other properties of the dot product? Well, A A = A x A x + A y A y = (A x ) 2 + (A y ) 2 = A 2, so the dot product of a vector with itself can be used to calculate the magnitude of vector A by taking the square root of the result. Also, when you take the dot product of the zero vector 0 and any vector, you get the scalar zero. To summarize: A B = B A A A = A 2 0 A = 0 Calculate the magnitude of vector V = 24i + 7j using the dot product. V 2 =V V = (V x ) 2 + (V y ) 2 = ( 24) = = 625. Therefore, the magnitude of V is just the square root of 625, which is 25. Vector V is 25 units long. Dot product using magnitudes and directions The definition we ve been using works very well when you have vectors A and B already in component form. However, if you instead have A and B in terms of magnitude and direction, an equivalent way of calculating the dot product is A B = A B cos θ,

3 Math 185 Vectors Page 3 where θ is the angle between vectors A and B. In science and engineering, we usually drop the absolute value bars and write A B = AB cos θ, where A and B are the magnitudes of vectors A and B, respectively, and θ is once again the angle between the two vectors. You can see once again from the fact that multiplying real numbers is commutative that the order of A and B in the dot product doesn t matter. A = 5 units at 50º B = 8 units at 110º The angle θ between these two vectors is (110º 50º) = 60º. Since cosine is an even function, it doesn t matter which angle comes first: cos 60º = cos 60º. Therefore, A B = AB cos θ = 5 8 cos 60º = 20. You can also tell from this definition that whether the dot product is positive or negative depends on the angle between the vectors. Since A B = AB cos θ, where A and B are the magnitudes of the vectors and are consequently always positive, the sign of A B will depend on the properties of cos θ. If θ is acute, cos θ will be positive. If θ is obtuse, then cos θ will be negative. Also, there are some special cases of angles: if θ = 0, cos θ = 1. If θ = 180º, then cos θ = 1. And lastly if θ = 90º (vectors are perpendicular), then the dot product will be exactly zero. This is an important point that bears repeating: for any vectors A and B, if A and B are perpendicular then the dot product A B is zero. The converse is also true: if A B is zero, then A and B are perpendicular. It is worth noting that vectors A and B do not have to have the same units. For example, when calculating the work done by a force F on an object moving through displacement d, the work may be calculated by W=F D = Fd cos θ. The units of force are newtons (N), the units of displacement are metres (m) and the units of work, a scalar, are joules (J). You may recall from your previous study of physics that if the force vector is perpendicular to the displacement vector, that the force is doing no work on the object. This agrees with our observation above that if θ = 90º, then the dot product will be zero.

4 Page 4 Math 185 Vectors Are A and B perpendicular? Use the dot product A B to determine your answer. A = 3i 5j B = 10i 6j A B = A x B x + A y B y = 3 ( 10) + ( 5) ( 6) = = 0. Yes, these vectors are perpendicular since A B = 0. The Angle Between Two Vectors Since A B = AB cos θ, you can solve for cos θ to get the following equation. cos AB AB This allows you to calculate the angle between the vectors from the magnitudes of the vectors and the dot product. Calculate the angle between the two vectors using the dot product. A = 3i 4j B = 2i 2j A B = A x B x + A y B y = 3 ( 2) + ( 4) ( 2) = = 2. A 2 = A A = (A x ) 2 + (A y ) 2 = (3) 2 + ( 4) 2 = = 25, so A = 5. B 2 = B B = (B x ) 2 + (B y ) 2 = ( 2) 2 + ( 2) 2 = = 8, so B = 8. Therefore, AB 2 2 cos. AB cos º. The angle between A and B is 81.9º.

5 Math 185 Vectors Page 5 Since the inverse of the cosine gives answers in quadrants I and II, you don t have to do any further calculations to get the angle. If the angle you are trying to calculate is obtuse (90º < θ < 180º), then you ll find that the dot product is negative and taking the inverse cosine will automatically give you an answer in QII. The Dot Product in Three Dimensions We can generalize to three dimensions: instead of restricting ourselves to x and y, we can add a third dimension by adding a z-axis which is perpendicular to both of the x- and y- axes. This means that the z-axis would be either pointing into or out of the page that the x- and y-axes are drawn on. Our convention is that we use a right-handed system. This means that, using your right hand, if your thumb is pointing along the x-direction and your fingers are pointing along the y-direction, that your palm points in the direction of the z-axis. For our usual x-y coordinate system drawn on a sheet of paper, this means that the z-axis is pointing out of the paper towards you. Points in this 3-dimensional space must therefore have three coordinates, not two, and are written as ordered triples: (x, y, z). Similarly, vectors will now have three components, such that vector A will have components A x, A y, and A z. Writing in ijk notation, we then have k, the unit vector pointing along the z-direction, associated with A z, so vector A could then be, for example, equal to 3i 4j + 5k. Three-dimensional vectors can also be written with magnitudes and directions. However, the direction angles are more complicated than we have time for, so we will restrict ourselves to calculating the magnitude of 3D vectors. We can now state the definition of the dot product in 3D form: A B = A x B x + A y B y + A z B z A = 3i 5j + 2k B = 2i 7j + k A B = A x B x + A y B y + A z B z = 3 ( 2) 5 ( 7) = = 31. As before, the dot product may be used to find the magnitude of a 3D vector, as in the following example.

6 Page 6 Math 185 Vectors Calculate the magnitude of vector V = 4i + 7j 3k using the dot product. V 2 =V V = (V x ) 2 + (V y ) 2 + (V z ) 2 = ( 4) ( 3) 2 = = 70. Therefore, the magnitude of V is just the square root of this dot product, 70. Vector V is 70 units long. We can also the dot product to find the angle between two 3D vectors in space, as in the following example. Calculate the angle between the two vectors using the dot product. A = 3i 5j + 2k B = 2i 7j + k A B = A x B x + A y B y + A z B z = 3 ( 2) 5 ( 7) = = 31. A 2 = A A = (A x ) 2 + (A y ) 2 + (A z ) 2 = (3) 2 + ( 5) 2 + (2) 2 = = 38, so A = 38. B 2 = B B = (B x ) 2 + (B y ) 2 + (B z ) 2 = ( 2) 2 + ( 7) 2 + (1) 2 = = 54, so B = AB 1 31 Therefore, cos cos AB = 46.8º. The angle between A and B is 46.8º.

### 9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes

The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

### Vector Math Computer Graphics Scott D. Anderson

Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about

### Vector Algebra II: Scalar and Vector Products

Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define

### Figure 1.1 Vector A and Vector F

CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

### Unified Lecture # 4 Vectors

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

### 6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

### Review A: Vector Analysis

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A-0 A.1 Vectors A-2 A.1.1 Introduction A-2 A.1.2 Properties of a Vector A-2 A.1.3 Application of Vectors

### Vectors Math 122 Calculus III D Joyce, Fall 2012

Vectors Math 122 Calculus III D Joyce, Fall 2012 Vectors in the plane R 2. A vector v can be interpreted as an arro in the plane R 2 ith a certain length and a certain direction. The same vector can be

### [1] Diagonal factorization

8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

### The Dot and Cross Products

The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and

### Introduction to Matrices

Introduction to Matrices Tom Davis tomrdavis@earthlinknet 1 Definitions A matrix (plural: matrices) is simply a rectangular array of things For now, we ll assume the things are numbers, but as you go on

### Vector has a magnitude and a direction. Scalar has a magnitude

Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude

### 9 Multiplication of Vectors: The Scalar or Dot Product

Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

### AP Physics - Vector Algrebra Tutorial

AP Physics - Vector Algrebra Tutorial Thomas Jefferson High School for Science and Technology AP Physics Team Summer 2013 1 CONTENTS CONTENTS Contents 1 Scalars and Vectors 3 2 Rectangular and Polar Form

### A vector is a directed line segment used to represent a vector quantity.

Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector

### Chapter 3 Vectors. m = m1 + m2 = 3 kg + 4 kg = 7 kg (3.1)

COROLLARY I. A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately. Isaac Newton - Principia

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

### Units, Physical Quantities, and Vectors

Chapter 1 Units, Physical Quantities, and Vectors PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 1 To learn

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### The Vector or Cross Product

The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero

### 13.4 THE CROSS PRODUCT

710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product

### Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi

Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in three-dimensional space, we also examine the

### Lecture 2 Matrix Operations

Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a

Bellwork a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.

### Mechanics 1: Vectors

Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized

### v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)

0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3-space. This time the outcome will be a vector in 3-space. Definition

### One advantage of this algebraic approach is that we can write down

. Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the x-axis points out

### PHYSICS 151 Notes for Online Lecture #6

PHYSICS 151 Notes for Online Lecture #6 Vectors - A vector is basically an arrow. The length of the arrow represents the magnitude (value) and the arrow points in the direction. Many different quantities

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### Section 10.4 Vectors

Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such

### Math 215 HW #6 Solutions

Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

### FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

### Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

### MAT 1341: REVIEW II SANGHOON BAEK

MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### Linear Algebra: Vectors

A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector

### 2 Session Two - Complex Numbers and Vectors

PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

### Two vectors are equal if they have the same length and direction. They do not

Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

### 13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

### Vectors and Scalars. AP Physics B

Vectors and Scalars P Physics Scalar SCLR is NY quantity in physics that has MGNITUDE, but NOT a direction associated with it. Magnitude numerical value with units. Scalar Example Speed Distance ge Magnitude

### In order to describe motion you need to describe the following properties.

Chapter 2 One Dimensional Kinematics How would you describe the following motion? Ex: random 1-D path speeding up and slowing down In order to describe motion you need to describe the following properties.

### Chapter 6. Work and Energy

Chapter 6 Work and Energy The concept of forces acting on a mass (one object) is intimately related to the concept of ENERGY production or storage. A mass accelerated to a non-zero speed carries energy

### The Force Table Introduction: Theory:

1 The Force Table Introduction: "The Force Table" is a simple tool for demonstrating Newton s First Law and the vector nature of forces. This tool is based on the principle of equilibrium. An object is

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Vectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.

Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type

### LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

### LINES AND PLANES IN R 3

LINES AND PLANES IN R 3 In this handout we will summarize the properties of the dot product and cross product and use them to present arious descriptions of lines and planes in three dimensional space.

### 6. LECTURE 6. Objectives

6. LECTURE 6 Objectives I understand how to use vectors to understand displacement. I can find the magnitude of a vector. I can sketch a vector. I can add and subtract vector. I can multiply a vector by

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### A Primer on Index Notation

A Primer on John Crimaldi August 28, 2006 1. Index versus Index notation (a.k.a. Cartesian notation) is a powerful tool for manipulating multidimensional equations. However, there are times when the more

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

### Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.

6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### LINES AND PLANES CHRIS JOHNSON

LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance

### Section 9.1 Vectors in Two Dimensions

Section 9.1 Vectors in Two Dimensions Geometric Description of Vectors A vector in the plane is a line segment with an assigned direction. We sketch a vector as shown in the first Figure below with an

### Torgerson s Classical MDS derivation: 1: Determining Coordinates from Euclidean Distances

Torgerson s Classical MDS derivation: 1: Determining Coordinates from Euclidean Distances It is possible to construct a matrix X of Cartesian coordinates of points in Euclidean space when we know the Euclidean

### Evaluating trigonometric functions

MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,

### 28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

### discuss how to describe points, lines and planes in 3 space.

Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position

### Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

### Physics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus

Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations

### Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

### Section 1: How will you be tested? This section will give you information about the different types of examination papers that are available.

REVISION CHECKLIST for IGCSE Mathematics 0580 A guide for students How to use this guide This guide describes what topics and skills you need to know for your IGCSE Mathematics examination. It will help

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### The Matrix Stiffness Method for 2D Trusses

The Matrix Stiffness Method for D Trusses Method CEE 4L. Matrix Structural Analysis Department of Civil and Environmental Engineering Duke University Henri P. Gavin Fall, 04. Number all of the nodes and

### 2.2 Magic with complex exponentials

2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### 8-3 Dot Products and Vector Projections

8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v

### Addition and Subtraction of Vectors

ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

### Section 1. Inequalities -5-4 -3-2 -1 0 1 2 3 4 5

Worksheet 2.4 Introduction to Inequalities Section 1 Inequalities The sign < stands for less than. It was introduced so that we could write in shorthand things like 3 is less than 5. This becomes 3 < 5.

### Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1

Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1 What are the multiples of 5? The multiples are in the five times table What are the factors of 90? Each of these is a pair of factors.

### The Matrix Elements of a 3 3 Orthogonal Matrix Revisited

Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, Three-Dimensional Proper and Improper Rotation Matrices, I provided a derivation

### Difference between a vector and a scalar quantity. N or 90 o. S or 270 o

Vectors Vectors and Scalars Distinguish between vector and scalar quantities, and give examples of each. method. A vector is represented in print by a bold italicized symbol, for example, F. A vector has

### Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

### TRIGONOMETRY Compound & Double angle formulae

TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae

### 3. KINEMATICS IN TWO DIMENSIONS; VECTORS.

3. KINEMATICS IN TWO DIMENSIONS; VECTORS. Key words: Motion in Two Dimensions, Scalars, Vectors, Addition of Vectors by Graphical Methods, Tail to Tip Method, Parallelogram Method, Negative Vector, Vector

### Objective. Materials. TI-73 Calculator

0. Objective To explore subtraction of integers using a number line. Activity 2 To develop strategies for subtracting integers. Materials TI-73 Calculator Integer Subtraction What s the Difference? Teacher

### Rotation Matrices and Homogeneous Transformations

Rotation Matrices and Homogeneous Transformations A coordinate frame in an n-dimensional space is defined by n mutually orthogonal unit vectors. In particular, for a two-dimensional (2D) space, i.e., n

### Figure 27.6b

Figure 27.6a Figure 27.6b Figure 27.6c Figure 27.25 Figure 27.13 When a charged particle moves through a magnetic field, the direction of the magnetic force on the particle at a certain point is A. in

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,

### Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

### MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:

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

### Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### Math Placement Test Practice Problems

Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

### Section 9.5: Equations of Lines and Planes

Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

### CHAPTER 5 Round-off errors

CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any

### Charlesworth School Year Group Maths Targets

Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve

### CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

### Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3