ex) What is the component form of the vector shown in the picture above?
|
|
- Mervyn Day
- 7 years ago
- Views:
Transcription
1 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 the placement of an arrow at the end of the line segment. ex) Pictured here is a ector (named) which has its initial point ( tail point) at ( 2, 1) and its terminal point (the arrow head ) at (2, 3). The component form of a ector is written using the form = ai+ bj. The alue of a is always written as a coefficient of i... which represents the ector s horizontal component. The alue of b is always written as a coefficient of j... which represents the ector s ertical component. ex) What is the component form of the ector shown in the picture aboe? The magnitude of a ector (which is denoted as ) is simply its length. It is calculated by applying the Pythagorean Theorem to the component coefficients a and b. 2 2 For any ector = ai+ bj, its magnitude is = a + b ex) What is the magnitude of the ector from the picture aboe? (Use the component form)
2 ex) Find the component form and the magnitude of the ector with initial point at ( 3, 11) and terminal point at(9, 40). (Approximate the magnitude to 2 decimal places.) The two main operations with ectors are ector addition and scalar multiplication. These operations can be done algebraically and graphically. Ex) For the ectors u= i+ 3j and = 2i j Plot ectors with their initial points at the origin and determine the following ector combinations. (a) u+ Find the sum algebraically (using component forms) Find the sum graphically (using parallelogram law) Calculate u+
3 (b) 2u+ 3 Find the sum algebraically (using component forms) Find the sum graphically (using parallelogram law) The 2 applied to u and the number 3 applied to are examples of scalar multiplication. The scalars will scale the length of each ector making them longer. Calculate 2u+ 3 (c) u Find the sum algebraically (using component forms) Find the sum graphically (using parallelogram law) Calculate u
4 Direction Angle for a Vector The direction angle is always considered to be the standard position angle starting on the positie x axis rotating counterclockwise to the ector s position. It can be determined by the formula: horizontal component tanθ = ertical component 1 To get the angle you ll need to use TAN on your calculator... BUT MAKE SURE YOUR ANGLE IS LOCATED IN THE CORRECT QUADRANT! ex) Determine the direction angle for the ector = 5i+ j. (IT ALWAYS HELPS TO SKETCH THE VECTOR FIRST) ex) Determine the direction angle for the ector = 4i+ 7j. 1 (SKETCH THE VECTOR FIRST AND BE CAREFUL USING TAN )
5 Unit Vectors When you want to presere the direction of a certain ector but apply a different length to it, you ll need to transform that ector in to a unit ector... essentially a ector of length 1. To get a unit ector, you diide the components by the magnitude: unit ector = ex) What is the unit ector which has the same direction as = 4i 3j? ex) A force of 1200 lbs is applied in the direction of the ector = 4i 3j. What are the components of this force ector? (Call the force ector F)
6 Decomposing a Vector When you already know the magnitude and direction angle, θ, of a ector, you can write it in component form using the formulas Horizontal Component cosθ Vertical Component sinθ Creating the ector = ( cos θ) i+ ( sin θ) j ex) Find the horizontal and ertical components of the ector with a length of = 800 and a direction ofθ = 145. (Round components to 2 decimal places). ex) A jet is flying in a direction of N 20 E with a speed of 500 mi/h. Represent the elocity of this jet as a ector in component form. (2 decimal place rounding)
7 Resultant Vectors The resultant of two or more ectors is the result of all of the ectors acting on the same object at the same time. A resultant is simply their ector sum. ex) Two tugboats are pulling a barge due north through a channel. One tugboat is pulling with a force of 3500 lbs at a heading of N 20 E and the other tugboat is pulling with 4000 lbs of force at a heading of N 25 W. (See the diagram.) a) What are the component forms of the force ector for each tugboat? b) Calculate the resultant ector. This is the ector sum of adding the tugboat ectors together. c) What is the magnitude and the direction of the resultant ector? Gie the direction as a bearing.
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
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 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 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 informationBasic Linear Algebra
Basic Linear Algebra by: Dan Sunday, softsurfer.com Table of Contents Coordinate Systems 1 Points and Vectors Basic Definitions Vector Addition Scalar Multiplication 3 Affine Addition 3 Vector Length 4
More informationModule 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 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 NON-CONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects
More informationSection 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 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 informationReview 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 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 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 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 x-axis points out
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 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 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 informationVector Algebra. Addition: (A + B) + C = A + (B + C) (associative) Subtraction: A B = A + (-B)
Vector Algebra When dealing with scalars, the usual math operations (+, -, ) are sufficient to obtain any information needed. When dealing with ectors, the magnitudes can be operated on as scalars, but
More informationLINES 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.
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 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 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 informationPlotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation
Plotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation ED 5661 Mathematics & Navigation Teacher Institute August 2011 By Serena Gay Target: Precalculus (grades 11 or 12) Lesson
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 informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More informationv 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 informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
More information13.4 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 information6. 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 informationsin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
More informationExample 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 informationArea 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 informationThe 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 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 informationMath, 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 informationChapter 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 informationVectors, velocity and displacement
Vectors, elocit and displacement Sample Modelling Actiities with Excel and Modellus ITforUS (Information Technolog for Understanding Science) 2007 IT for US - The project is funded with support from the
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 informationChapter 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 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 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 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 informationLecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.8-4.12, second half of section 4.7
Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.8-4.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal
More 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 three-space, we write a vector in terms
More informationCHAPTER ONE VECTOR GEOMETRY
CHAPTER ONE VECTOR GEOMETRY. INTRODUCTION In this chapter ectors are first introdced as geometric objects, namely as directed line segments, or arrows. The operations of addition, sbtraction, and mltiplication
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 informationVECTOR 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 informationVectors 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 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 informationIntroduction and Mathematical Concepts
CHAPTER 1 Introduction and Mathematical Concepts PREVIEW In this chapter you will be introduced to the physical units most frequently encountered in physics. After completion of the chapter you will be
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 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 informationANALYTICAL 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 informationUse Geometry Expressions to find equations of curves. Use Geometry Expressions to translate and dilate figures.
Learning Objectives Loci and Conics Lesson 2: The Circle Level: Precalculus Time required: 90 minutes Students are now acquainted with the idea of locus, and how Geometry Expressions can be used to explore
More informationAddition 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 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 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 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 informationChapter 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 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 8:15-10:15 Room:
More informationWorksheet to Review Vector and Scalar Properties
Worksheet to Review Vector and Scalar Properties 1. Differentiate between vectors and scalar quantities 2. Know what is being requested when the question asks for the magnitude of a quantity 3. Define
More informationVectors 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 informationMAC 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 informationHow To Solve The Pythagorean Triangle
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
More informationThe 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 informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationDr. Fritz Wilhelm, DVC,8/30/2004;4:25 PM E:\Excel files\ch 03 Vector calculations.doc Last printed 8/30/2004 4:25:00 PM
E:\Ecel files\ch 03 Vector calculations.doc Last printed 8/30/2004 4:25:00 PM Vector calculations 1 of 6 Vectors are ordered sequences of numbers. In three dimensions we write vectors in an of the following
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
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 informationSOLID MECHANICS DYNAMICS TUTORIAL CENTRIPETAL FORCE
SOLID MECHANICS DYNAMICS TUTORIAL CENTRIPETAL FORCE This work coers elements of the syllabus for the Engineering Council Exam D5 Dynamics of Mechanical Systems C10 Engineering Science. This tutorial examines
More informationCOMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i
COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More informationSECTION 7-4 Algebraic Vectors
7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More informationAdding 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 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 informationVector 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 informationBASIC MATHEMATICS FOR ELECTRICITY
MEBA Engineering School BASIC MATHEMATICS FOR ELECTRICITY (Revised: January 2002) This book belongs to: math\cover.vsd (yellow) 01/01/2002 This page is intentionally left blank. Preface Most of this material
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 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 informationDear Accelerated Pre-Calculus Student:
Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also
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 information6. 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 informationRotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012
Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to
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 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 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 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 informationChapter 4. Forces and Newton s Laws of Motion. continued
Chapter 4 Forces and Newton s Laws of Motion continued 4.9 Static and Kinetic Frictional Forces When an object is in contact with a surface forces can act on the objects. The component of this force acting
More information4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles
4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred
More informationwith functions, expressions and equations which follow in units 3 and 4.
Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More information... ... . (2,4,5).. ...
12 Three Dimensions ½¾º½ Ì ÓÓÖ Ò Ø ËÝ Ø Ñ So far wehave been investigatingfunctions ofthe form y = f(x), withone independent and one dependent variable Such functions can be represented in two dimensions,
More informationPhysics 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 informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationPart I. Basic Maths for Game Design
Part I Basic Maths for Game Design 1 Chapter 1 Basic Vector Algebra 1.1 What's a vector? Why do you need it? A vector is a mathematical object used to represent some magnitudes. For example, temperature
More informationWeight The weight of an object is defined as the gravitational force acting on the object. Unit: Newton (N)
Gravitational Field A gravitational field as a region in which an object experiences a force due to gravitational attraction Gravitational Field Strength The gravitational field strength at a point in
More informationGeorgia Standards of Excellence Frameworks. Mathematics. GSE Pre-Calculus Unit 7: Vectors
Georgia Standards of Excellence Frameworks Mathematics GSE Pre-Calculus Unit 7: Vectors These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.
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 informationNational 5 Mathematics Course Assessment Specification (C747 75)
National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for
More informationHomework 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 informationMathematics 205 HWK 6 Solutions Section 13.3 p627. Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors.
Mathematics 205 HWK 6 Solutions Section 13.3 p627 Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors. Problem 5, 13.3, p627. Given a = 2j + k or a = (0,2,
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
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 information