Section 9.1 Vectors in Two Dimensions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Section 9.1 Vectors in Two Dimensions"

Transcription

1 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 arrow to specify the direction. We denote this vector by AB. Point A is the initial point, and B is the terminal point of the vector AB. The length of the line segment AB is called the magnitude or length of the vector and is denoted by AB. We use boldface letters to denote vectors. Thus, we write u = AB. Two vectors are considered equal if they have equal magnitude and the same direction. The Figures below illustrate the way we (a) add (b) subtract (c) multiply a vector by a scalar 1

2 Vectors in the Coordinate Plane By placing a vector in a coordinate plane, we can describe it analytically (that is, by using components). We represent a vector v as an ordered pair of real numbers. v = a,b where a is the horizontal component of v and b is the vertical component of v. The vector a, b has many different representations, depending on its initial point. (a) Find the component form of the vector u with initial point ( 2,5) and terminal point (,7). (b) If the vector v =,7 is sketched with initial point (2,4), what is its terminal point? (c) Sketch representations of the vector w = 2, with initial points at (0,0), (2,2), ( 2, 1), and (1,4). (a) The desired vector is u = ( 2),7 5 = 5,2 (b) Let the terminal point of v be (x,y). Then x 2,y 4 =,7 So x 2 = and y 4 = 7, or x = 5 and y = 11. The terminal point is (5,11). (c) Representations of the vector w are sketched in the Figure above. (a) Find the vector u with initial point ( 5,4) and terminal point (4,8). (b) If the vector v = 4,7 is sketched with initial point (4,5), what is its terminal point? 2

3 (a) Find the vector u with initial point ( 5,4) and terminal point (4,8). (b) If the vector v = 4,7 is sketched with initial point (4,5), what is its terminal point? (a) The desired vector is u = 4 ( 5),8 4 = 9,4 (b) Let the terminal point of v be (x,y). Then x 4,y 5 = 4,7 So x 4 = 4 and y 5 = 7, or x = 8 and y = 12. The terminal point is (8,12). Find the magnitude of each vector. (a) u = 2, (b) v = 5,0 (c) w = 5, 4 5 (a) u = 2 2 +( ) 2 = 1 (b) v = = 25 = 5 ( ) 2 ( ) (c) w = + = = 25 = 1 If u = 2, and v = 1,2, find u+v, u v, 2u, v, and 2u+v.

4 If u = 2, and v = 1,2, find u+v, u v, 2u, v, and 2u+v. We have u+v = 2, + 1,2 = 2+( 1), +2 = 1, 1 u v = 2, 1,2 = 2 ( 1), 2 =, 5 2u = 2 2, = 2(2),2( ) = 4, 6 v = 1,2 = ( )( 1),( )2 =, 6 2u+v = 2 2, + 1,2 = 4, 6 +,6 = 4+( ), 6+6 = 1,0 The zero vector is the vector 0 = 0,0. A vector of length 1 is called a unit vector. For instance, in the Example above, the vector w = 5, 4 is a unit vector. Two useful unit 5 vectors are i and j, defined by i = 1,0 j = 0,1 These vectors are special because any vector can be expressed in terms of them. (a) Write the vector u = 5, 8 in terms of i and j. (b) If u = i+2j and v = i+6j, write 2u+5v in terms of i and j. 4

5 (a) Write the vector u = 5, 8 in terms of i and j. (b) If u = i+2j and v = i+6j, write 2u+5v in terms of i and j. (a) u = 5i+( 8)j = 5i 8j (b) The properties of addition and scalar multiplication of vectors show that we can manipulate vectors in the same way as algebraic expressions. Thus 2u+5v = 2(i+2j)+5( i+6j) = (6i+4j)+( 5i+0j) = (6 5)i+(4+0)j = i+4j (a) A vector v has length 8 and direction π/. Find the horizontal and vertical components, and write v in terms of i and j. (b) Find the direction of the vector u = i+j. (a) We have v = a,b, where the components are given by a = 8cos π = = 4 and b = 8sin π = 8 Thus, v = 4,4 = 4i+4 j. 2 = 4 (b) From the Figure on the right we see that the direction θ has the property that tanθ = 1 = Thus, the reference angle for θ is π/6. Since the terminal point of the vector u is in Quadrant II, it follows that θ = 5π/6. A woman launches a boat from one shore of a straight river and wants to land at the point directly on the opposite shore. If the speed of the boat (relative to the water) is 10 mi/h and the river is flowing east at the rate of 5 mi/h, in what direction should she head the boat in order to arrive at the desired landing point? 5

6 A woman launches a boat from one shore of a straight river and wants to land at the point directly on the opposite shore. If the speed of the boat (relative to the water) is 10 mi/h and the river is flowing east at the rate of 5 mi/h, in what direction should she head the boat in order to arrive at the desired landing point? We choose a coordinate system with the origin at the initial position of the boat as shown in the Figure above. Let u and v represent the velocities of the river and the boat, respectively. Clearly, u = 5i, and since the speed of the boat is 10 mi/h, we have v = 10, so v = (10cosθ)i+(10sinθ)j where the angle θ is as shown in the Figure above. The true course of the boat is given by the vector w = u+v. We have w = u+v = 5i+(10cosθ)i+(10sinθ)j = (5+10cosθ)i+(10sinθ)j Since the woman wants to land at a point directly across the river, her direction should have horizontal component 0. In other words, she should choose θ in such a way that 5+10cosθ = 0 cosθ = 1 2 θ = 120 Thus she should head the boat in the direction θ = 120 (or N 0 W). 6

ex) What is the component form of the vector shown in the picture above?

ex) What is the component form of the vector shown in the picture above? Vectors A ector is a directed line segment, which has both a magnitude (length) and direction. A ector can be created using any two points in the plane, the direction of the ector is usually denoted by

More information

Figure 1.1 Vector A and Vector F

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

More information

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

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

More information

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

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.

More information

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

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

More information

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

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

More information

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

More information

Review A: Vector Analysis

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

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

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

More information

Unified Lecture # 4 Vectors

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,

More information

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

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

More information

5.3 The Cross Product in R 3

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

More information

Linear Algebra: Vectors

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

More information

Addition and Subtraction of Vectors

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

More information

Unit 11 Additional Topics in Trigonometry - Classwork

Unit 11 Additional Topics in Trigonometry - Classwork Unit 11 Additional Topics in Trigonometry - Classwork In geometry and physics, concepts such as temperature, mass, time, length, area, and volume can be quantified with a single real number. These are

More information

Section 10.4 Vectors

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

More information

AP Physics - Vector Algrebra Tutorial

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

More information

Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.2: Magnitudes, Directions, and Components of Vectors Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

More information

Mechanics 1: Vectors

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

More information

Vectors. Vector Multiplication

Vectors. Vector Multiplication Vectors Directed Line Segments and Geometric Vectors A line segment to which a direction has been assigned is called a directed line segment. The figure below shows a directed line segment form P to Q.

More information

Vectors and Scalars. AP Physics B

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

More information

The Vector or Cross Product

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

More information

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

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

More information

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. 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

More information

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 1 NON-CONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects

More information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

More information

TEACHER ANSWER KEY November 12, 2003. Phys - Vectors 11-13-2003

TEACHER ANSWER KEY November 12, 2003. Phys - Vectors 11-13-2003 Phys - Vectors 11-13-2003 TEACHER ANSWER KEY November 12, 2003 5 1. A 1.5-kilogram lab cart is accelerated uniformly from rest to a speed of 2.0 meters per second in 0.50 second. What is the magnitude

More information

Vector Spaces; the Space R n

Vector Spaces; the Space R n Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which

More information

Examples of Scalar and Vector Quantities 1. Candidates should be able to : QUANTITY VECTOR SCALAR

Examples of Scalar and Vector Quantities 1. Candidates should be able to : QUANTITY VECTOR SCALAR Candidates should be able to : Examples of Scalar and Vector Quantities 1 QUANTITY VECTOR SCALAR Define scalar and vector quantities and give examples. Draw and use a vector triangle to determine the resultant

More information

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

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,

More information

Homework 2 Solutions

Homework 2 Solutions Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to

More information

PHYSICS 151 Notes for Online Lecture #6

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

More information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space 11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

More information

Triangle Trigonometry and Circles

Triangle Trigonometry and Circles Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of

More information

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)

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

More information

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. 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

More information

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.

More information

Vector Algebra II: Scalar and Vector Products

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

More information

6. LECTURE 6. Objectives

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

More information

Exam 1 Review Questions PHY 2425 - Exam 1

Exam 1 Review Questions PHY 2425 - Exam 1 Exam 1 Review Questions PHY 2425 - Exam 1 Exam 1H Rev Ques.doc - 1 - Section: 1 7 Topic: General Properties of Vectors Type: Conceptual 1 Given vector A, the vector 3 A A) has a magnitude 3 times that

More information

9 Multiplication of Vectors: The Scalar or Dot Product

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

More information

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

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

More information

3. KINEMATICS IN TWO DIMENSIONS; VECTORS.

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

More information

2.2 Magic with complex exponentials

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

More information

Section V.3: Dot Product

Section V.3: Dot Product 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,

More information

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1 Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

More information

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

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

More information

VECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a.

VECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a. VECTOR ALGEBRA Chapter 10 101 Overview 1011 A quantity that has magnitude as well as direction is called a vector 101 The unit vector in the direction of a a is given y a and is represented y a 101 Position

More information

Chapter 3B - Vectors. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 3B - Vectors. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University Chapter 3B - Vectors A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007 Vectors Surveyors use accurate measures of magnitudes and directions to

More information

Lab 2: Vector Analysis

Lab 2: Vector Analysis Lab 2: Vector Analysis Objectives: to practice using graphical and analytical methods to add vectors in two dimensions Equipment: Meter stick Ruler Protractor Force table Ring Pulleys with attachments

More information

Vectors Math 122 Calculus III D Joyce, Fall 2012

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

More information

Module 8 Lesson 4: Applications of Vectors

Module 8 Lesson 4: Applications of Vectors Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems

More information

(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its

(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its (1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:

More information

Vectors VECTOR PRODUCT. Graham S McDonald. A Tutorial Module for learning about the vector product of two vectors. Table of contents Begin Tutorial

Vectors VECTOR PRODUCT. Graham S McDonald. A Tutorial Module for learning about the vector product of two vectors. Table of contents Begin Tutorial Vectors VECTOR PRODUCT Graham S McDonald A Tutorial Module for learning about the vector product of two vectors Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk 1. Theory 2. Exercises

More information

The Matrix Elements of a 3 3 Orthogonal Matrix Revisited

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

More information

Rotation Matrices and Homogeneous Transformations

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

More information

2 Session Two - Complex Numbers and Vectors

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

More information

2. Spin Chemistry and the Vector Model

2. Spin Chemistry and the Vector Model 2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing

More information

Chapter 3 Practice Test

Chapter 3 Practice Test Chapter 3 Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which of the following is a physical quantity that has both magnitude and direction?

More information

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

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

More information

13.4 THE CROSS PRODUCT

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

More information

Math, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.

Math, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image. Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical

More information

F B = ilbsin(f), L x B because we take current i to be a positive quantity. The force FB. L and. B as shown in the Figure below.

F B = ilbsin(f), L x B because we take current i to be a positive quantity. The force FB. L and. B as shown in the Figure below. PHYSICS 176 UNIVERSITY PHYSICS LAB II Experiment 9 Magnetic Force on a Current Carrying Wire Equipment: Supplies: Unit. Electronic balance, Power supply, Ammeter, Lab stand Current Loop PC Boards, Magnet

More information

Largest Fixed-Aspect, Axis-Aligned Rectangle

Largest Fixed-Aspect, Axis-Aligned Rectangle Largest Fixed-Aspect, Axis-Aligned Rectangle David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created: February 21, 2004 Last Modified: February

More information

Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

More information

Lecture 3: Coordinate Systems and Transformations

Lecture 3: Coordinate Systems and Transformations Lecture 3: Coordinate Systems and Transformations Topics: 1. Coordinate systems and frames 2. Change of frames 3. Affine transformations 4. Rotation, translation, scaling, and shear 5. Rotation about an

More information

FURTHER VECTORS (MEI)

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

More information

Math 241, Exam 1 Information.

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)

More information

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

More information

Section 9.5: Equations of Lines and Planes

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

More information

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

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

More information

Concepts in Calculus III

Concepts in Calculus III Concepts in Calculus III Beta Version UNIVERSITY PRESS OF FLORIDA Florida A&M University, Tallahassee Florida Atlantic University, Boca Raton Florida Gulf Coast University, Ft. Myers Florida International

More information

Vector Algebra CHAPTER 13. Ü13.1. Basic Concepts

Vector Algebra CHAPTER 13. Ü13.1. Basic Concepts CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have

More information

Area and Arc Length in Polar Coordinates

Area and Arc Length in Polar Coordinates Area and Arc Length in Polar Coordinates The Cartesian Coordinate System (rectangular coordinates) is not always the most convenient way to describe points, or relations in the plane. There are certainly

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

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

More information

Vector Math Computer Graphics Scott D. Anderson

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

More information

The Force Table Introduction: Theory:

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

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

More information

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.

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

More information

Section 1.1. Introduction to R n

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

More information

Web review - Ch 3 motion in two dimensions practice test

Web review - Ch 3 motion in two dimensions practice test Name: Class: _ Date: _ Web review - Ch 3 motion in two dimensions practice test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which type of quantity

More information

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

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

More information

Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER

More information

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col

More information

Mathematics Notes for Class 12 chapter 10. Vector Algebra

Mathematics Notes for Class 12 chapter 10. Vector Algebra 1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative

More information

Geometry and Measurement

Geometry and Measurement The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

More information

Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi

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

More information

BALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle

BALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and

More information

Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

More information

Physics 590 Homework, Week 6 Week 6, Homework 1

Physics 590 Homework, Week 6 Week 6, Homework 1 Physics 590 Homework, Week 6 Week 6, Homework 1 Prob. 6.1.1 A descent vehicle landing on the moon has a vertical velocity toward the surface of the moon of 35 m/s. At the same time it has a horizontal

More information

MAT 1341: REVIEW II SANGHOON BAEK

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

More information

Equations Involving Lines and Planes Standard equations for lines in space

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

More information

A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form

A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...

More information

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 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

More information

1. Units of a magnetic field might be: A. C m/s B. C s/m C. C/kg D. kg/c s E. N/C m ans: D

1. Units of a magnetic field might be: A. C m/s B. C s/m C. C/kg D. kg/c s E. N/C m ans: D Chapter 28: MAGNETIC FIELDS 1 Units of a magnetic field might be: A C m/s B C s/m C C/kg D kg/c s E N/C m 2 In the formula F = q v B: A F must be perpendicular to v but not necessarily to B B F must be

More information

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

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

More information

All About Motion - Displacement, Velocity and Acceleration

All About Motion - Displacement, Velocity and Acceleration All About Motion - Displacement, Velocity and Acceleration Program Synopsis 2008 20 minutes Teacher Notes: Ian Walter Dip App Chem; GDipEd Admin; TTTC This program explores vector and scalar quantities

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

The Force Table Vector Addition and Resolution

The Force Table Vector Addition and Resolution Name School Date The Force Table Vector Addition and Resolution Vectors? I don't have any vectors, I'm just a kid. From Flight of the Navigator Explore the Apparatus/Theory We ll use the Force Table Apparatus

More information

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

More information