Section 10.4 Vectors


 Anabel Washington
 3 years ago
 Views:
Transcription
1 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 as v), but since we can't write in bold, we put an arrow over the letter that represents a vector (such as v or v ). When writing, I will use the latter notation (like the top half of a horizontal arrow) because it is the quickest to write and also because this is the notation that physicists typically use, and we learn about vectors mainly for use in future physics classes. A vector has both a magnitude (length) and direction. The arrowhead is always at the terminal point of the vector and it shows you which direction a vector has traveled. Two vectors are equal if they have the same magnitude and direction, even if they don't have the same initial and terminating points. All three vectors in the figure to the right are equal to one another. Two vectors that have the same magnitude but exactly opposite directions are denoted as v and v. When we talk about the direction of a vector we are really talking about two things: the angle the vector makes (you might think of this as its slope) and also whether it is pointing up, down, left, or right. To have "opposite directions" the slope must be the same (so the vectors must be ) but the pointing direction must be opposite. Let's look at some examples. In the top box of the figure to the left, both vectors are clearly the same length, so they have the same magnitude. Additionally, the vectors are parallel to one another, so they have the same slope. One is pointing right while the other is pointing left. So these are opposite vectors. In the second box, the vectors have the same magnitude and the same slope. One is pointing up and the other is pointing down. So these are opposite vectors also. In the last box, the vectors have the same magnitude, but they are not parallel. Because they do not have the same slope, these are not opposite vectors. ADDING VECTORS To add vectors v w, you place the initial point of w at the terminal point of v. Then the resultant sum vector v w is the vector that takes you from the initial point of v to the terminal point of w, as shown in the figure to the right. The order you add vectors in does not matter, so the vector v w is the same as the vector w v. Try this in the figure to the right. Draw w, then add v to it. Notice that the vector you get has the exact same magnitude and direction as the vector v w does. This means that vector addition is commutative Page 1
2 Vector addition is also associative, as shown in the figure to the right. Whether you add u v then w, or v w then u, or u w then v (not shown), the resultant vector will still have the same magnitude and direction. The zero vector, 0, has the property that v 0 0 v v. Also, if you add a vector, v, to its opposite vector, v, you get the zero vector: v v 0. The difference of two vectors is defined as follows: v w v w. So we never actually subtract vectors; instead, we add the opposite of the second vector. v and w are shown in the figure to the left. To find the difference v w, we add w to the terminal point of v. The resultant vector v w starts at the initial point of v and ends at the terminal point of w. MULTIPLYING VECTORS BY NUMBERS In the world of vectors, we call real numbers scalars. Scalars have only magnitude; they do not have direction. If we multiply a positive scalar () times a vector (v ), the magnitude becomes times the magnitude of the vector, but the direction is unchanged. If we multiply a negative scalar () times a vector (v ), the magnitude becomes times the magnitude of the vector and the direction changes to the opposite of v. Draw the vector 2v in the figure above. GRAPHING VECTORS Example: Given the vectors u, v, and w shown in the lefthand box below, graph the following: a) v w b) u v c) 2w d) 3v e) u 2w v u w v MAGNITUDES OF VECTORS AND THE UNIT VECTOR As we've already learned, the magnitude of a vector is its length. We will use the notation v to represent the magnitude of v. A magnitude is always positive (since there is no such thing as a negative length). A vector u that has a magnitude of 1 u 1 is called a unit vector Page 2
3 FINDING A POSITION VECTOR In order to graph the vectors in the previous example, we had to count how many units right or left and up or down a vector traveled. For instance, from its initial point to its terminal point, the vector w moved right 3 and down 2. If we assumed that the initial point was at the origin, then the terminal point would be at. But because this is a vector, we use the notation v a, b, where a and b are called the components of the vector. A vector written in this manner v a, b is called an algebraic vector. The specific case where a vector has its initial point at the origin is called a position vector. Recall that a unit vector has a magnitude of. If we say that i is the unit vector that points directly along the xaxis and j is the unit vector that points directly along the yaxis (as shown in the figure to the right), then we can rewrite v a, b as v ai bj, which indicates we move a units to the right or left and b units up or down. Label the vectors u, v, and w from the previous page in their component form, either a, b or ai bj. u v w A vector whose initial point is not at the origin can be rewritten algebraically to find an equivalent position vector by subtracting the components of the initial point from the components of the terminal point. Example: The vector v has initial point P and terminal point Q. Write v in the form ai bj (i.e. find its position vector). a) P = (3, 2); Q = (6, 5) (b) P = (1, 4); Q = (6, 2) v 6 3,5 2 9,3 v 9i 3j (c) P = (1, 1); Q = (2, 2) The magnitude of a vector v ai bj can be found using the formula v a b. Example: Find v. a) v 5i 12 j b) v i j 10.4 Page 3
4 ADDING AND SUBTRACTING VECTORS ALGEBRAICALLY Addition and subtraction of vectors is very straightforward: you add or subtract the like components together (i's with i's, j's with j's). To multiply a scalar by a vector, you multiply the scalar by each component of the vector (you can think of it as distributing). Example: If v 3i 5j and w 2i 3 j, find the following: a) 3v 2w 3 3i 5j i 3 j 9i 15j 4i 6 j 9 4 i 15 6 j 13i 21 j or 13, 21 b) 4v 3w c) v w a1, b2 i j i j i j i j d) v w FINDING A UNIT VECTOR In some applications it will be helpful to be able to find a unit vector (remember a unit vector has a magnitude of 1) that v has the same direction as a given vector. To do this, you simply divide a vector by its magnitude. So u. v Example: Find the unit vector in the same direction as v. a) v 5i 12 j b) v 2i j v Then 5 12 u i j Page 4
5 FINDING A VECTOR FROM ITS MAGNITUDE AND DIRECTION In many applications, a vector is described by its magnitude (often a speed or force) and direction (an angle, ) rather than its i and j components. In these cases it is necessary to determine the algebraic form of the vector using the formula: Example: Write the vector v in the form ai bj, given its magnitude v and the angle it makes with the positive xaxis. a) v 8, 45 2 Distribute Since cos45 sin45, we have v 8 i j v 4 2 i 4 2 j 2 b) v 3, 240 ANALYZING OBJECTS IN STATIC EQUILIBRIUM Forces can be represented by vectors, and when two forces act simultaneously on an object, their components add together to create a resultant force F1 F2. An object is said to be in static equilibrium if the object is at rest and the sum of all forces acting on the object is zero. Example: #68) A weight of 800 pounds is suspended from two cables. The left cable makes an angle of 35 and the right cable makes an angle of 50 with the beam. What are the tensions in the two cables? F pounds F 3 F 2 Assuming the weight is stationary, there are three forces creating static equilibrium: the force (tension) of each cable F1 and F2, and the force of the weight itself F 3. We need to determine the algebraic components of F1 and F2. We already know the magnitude (force) of the weight itself: it has a force of 800 pounds, being pulled straight down (due to gravity), so its force can be written as F3 800 j. We need to find the angle that each vector makes with the positive xaxis so we can plug them into the formula F F cos i sin j. Assuming the point where the weight attaches to the cables is the origin, label the angles in the figure above. 1 = 2 = 10.4 Page 5
6 Example (continued): We do not know the magnitude of the force (tension) in each cable (this is what the problem asks us to find). So we will leave them as F 1 and F 2 in the formulas. We use a calculator to find decimal approximations of cosine and sine. F F cos145i sin145 j F i j F F i F j F F cos50i sin50 j F i j F F i F j Because these three forces are in static equilibrium, their sum must equal zero. This means that the sum of each component must equal zero. F F F F F i F F 800 j We start by finding a relationship between F 1 and F 2 using the i component: i component: F F F F F F We plug this result in for F 2 when working with the j component: j component: F F F F Since F F F F F 800 F lb F lb, then F F 2 Example: Repeat the previous problem if the lefthand cable is attached to the beam at a 30 angle, the righthand cable is attached to the beam at a 55 angle, and the weight being suspended is 950 pounds Page 6
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 information6. Vectors. 1 20092016 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 informationOne 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 xaxis points out
More informationSection 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 informationFigure 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 informationv w is orthogonal to both v and w. the three vectors v, w and v w form a righthanded 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 informationSection 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
More informationShear Force and Moment Diagrams
C h a p t e r 9 Shear Force and Moment Diagrams In this chapter, you will learn the following to World Class standards: Making a Shear Force Diagram Simple Shear Force Diagram Practice Problems More Complex
More informationex) 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 informationA 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 informationLecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.84.12, second half of section 4.7
Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.84.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationVector 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 informationDifference 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 informationUnit 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 informationIntroduction to Structural Mechanics 11 Introduction
Introduction to Structural Mechanics 11 Introduction In an effort to compete with film and TV, theatrical stage scenery has been growing larger, more complicated and more ambitions year after year. This
More informationGRAPHING IN POLAR COORDINATES SYMMETRY
GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry  yaxis,
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationEDEXCEL 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 NONCONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects
More informationPlot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.
Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line
More informationSection 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 information11.1. 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 informationPHYSICS 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 informationSolving 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 information9 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 informationElements of a graph. Click on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and yintercept in the equation of a line Comparing lines on
More informationVector 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 informationUnits, 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
More informationa.) 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.
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More informationDefinition: 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 informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationTrigonometric 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
More informationUnified 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 information2 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 informationPhysics Midterm Review Packet January 2010
Physics Midterm Review Packet January 2010 This Packet is a Study Guide, not a replacement for studying from your notes, tests, quizzes, and textbook. Midterm Date: Thursday, January 28 th 8:1510:15 Room:
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3space, as well as define the angle between two nonparallel planes, and determine the distance
More informationEvaluating trigonometric functions
MATH 1110 0090906 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
More informationSECTION 74 Algebraic Vectors
74 lgebraic Vectors 531 SECTIN 74 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More informationChapter 18 Static Equilibrium
Chapter 8 Static Equilibrium 8. Introduction Static Equilibrium... 8. Lever Law... Example 8. Lever Law... 4 8.3 Generalized Lever Law... 5 8.4 Worked Examples... 7 Example 8. Suspended Rod... 7 Example
More informationReview A: Vector Analysis
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A0 A.1 Vectors A2 A.1.1 Introduction A2 A.1.2 Properties of a Vector A2 A.1.3 Application of Vectors
More informationBiggar 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
More informationSection 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 informationVector 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 informationBasic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 52 = 72 5 + (22) = 72 5 = 5. x + 55 = 75. x + 0 = 20.
Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM 1. Introduction (really easy) An equation represents the equivalence between two quantities. The two sides of the equation are in balance, and solving
More information3. 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 informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationThe 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
More informationVectors. 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 informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra  Linear Equations & Inequalities T37/H37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationFRICTION, WORK, AND THE INCLINED PLANE
FRICTION, WORK, AND THE INCLINED PLANE Objective: To measure the coefficient of static and inetic friction between a bloc and an inclined plane and to examine the relationship between the plane s angle
More informationChapter 11 Equilibrium
11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationPROBLEM 2.9. sin 75 sin 65. R = 665 lb. sin 75 sin 40
POBLEM 2.9 A telephone cable is clamped at A to the pole AB. Knowing that the tension in the righthand portion of the cable is T 2 1000 lb, determine b trigonometr (a) the required tension T 1 in the
More information9.4. 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 socalled because when the scalar product of
More informationTwo 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
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationVector 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 informationLecture 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 informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More informationThe 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 information83 Dot Products and Vector Projections
83 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
More informationVectors 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 informationCross 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 information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationFURTHER 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: 97 Mathematics
More informationa cos x + b sin x = R cos(x α)
a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this
More informationWork Energy & Power. September 2000 Number 05. 1. Work If a force acts on a body and causes it to move, then the force is doing work.
PhysicsFactsheet September 2000 Number 05 Work Energy & Power 1. Work If a force acts on a body and causes it to move, then the force is doing work. W = Fs W = work done (J) F = force applied (N) s = distance
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More information5.3 SOLVING TRIGONOMETRIC EQUATIONS. Copyright Cengage Learning. All rights reserved.
5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations
More informationExamples 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 information13 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 threespace, we write a vector in terms
More informationDetermine If An Equation Represents a Function
Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The
More informationAlgebra. 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
More informationv v ax v a x a v a v = = = Since F = ma, it follows that a = F/m. The mass of the arrow is unchanged, and ( )
Week 3 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationAP 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 informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationAP PHYSICS C Mechanics  SUMMER ASSIGNMENT FOR 20162017
AP PHYSICS C Mechanics  SUMMER ASSIGNMENT FOR 20162017 Dear Student: The AP physics course you have signed up for is designed to prepare you for a superior performance on the AP test. To complete material
More informationTemperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.
Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is
More information28 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 informationEL9650/9600c/9450/9400 Handbook Vol. 1
Graphing Calculator EL9650/9600c/9450/9400 Handbook Vol. Algebra EL9650 EL9450 Contents. Linear Equations  Slope and Intercept of Linear Equations 2 Parallel and Perpendicular Lines 2. Quadratic Equations
More informationAP Physics Applying Forces
AP Physics Applying Forces This section of your text will be very tedious, very tedious indeed. (The Physics Kahuna is just as sorry as he can be.) It s mostly just a bunch of complicated problems and
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationCHAPTER 4 DIMENSIONAL ANALYSIS
CHAPTER 4 DIMENSIONAL ANALYSIS 1. DIMENSIONAL ANALYSIS Dimensional analysis, which is also known as the factor label method or unit conversion method, is an extremely important tool in the field of chemistry.
More informationLab 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 informationUniversal Law of Gravitation
Universal Law of Gravitation Law: Every body exerts a force of attraction on every other body. This force called, gravity, is relatively weak and decreases rapidly with the distance separating the bodies
More information1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.
1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationEquations 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 information2.2 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 informationMath 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
More informationF 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 information5.3 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 informationMagnetic Fields and Their Effects
Name Date Time to Complete h m Partner Course/ Section / Grade Magnetic Fields and Their Effects This experiment is intended to give you some handson experience with the effects of, and in some cases
More informationCh 7 Kinetic Energy and Work. Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43
Ch 7 Kinetic Energy and Work Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43 Technical definition of energy a scalar quantity that is associated with that state of one or more objects The state
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (1,3), (3,3), (2,3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the xcomponent of a point in the form (x,y). Range refers to the set of possible values of the ycomponent of a point in
More information