COMPONENTS OF VECTORS
|
|
- Beverly Washington
- 8 years ago
- Views:
Transcription
1 COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two perpendicular vectors A = A + A where A is parallel to the ais while A is parallel to the ais, for eample A A A A (1) A Since the A vector is alwas parallel to the ais, we ma describe it b a single signed number A, which is positive when A points right but negative when A points left. Likewise, the A vectors ma be described b a single signed number A positive when A points up but negative when A points down. The two signed numbers A and A are called the componentsof the vector A. In two dimensions, an vector V can be completel specified b its components (V,V ). Describing motion in all 3 dimensions of space requires a coordinate sstem with 3 perpendicular aes (,,z). Consequentl, a 3D vector V has three components (V,V,V z ), and we need to know all 3 components to completel specif the vector. 1
2 Adding vectors in components. In component notations, adding vectors is ver eas: The components of a vector sum C = A+ B are simpl the algebraic sums C = A + B, C = A + B. (2) Here is the geometric eplanation of this rule: C B C B B (3) C A A A In the same wa, we ma sum up several vectors: To get the components of a vector sum C = A 1 + A 2 + A A n, (4) we separatel sum up the components of all the vectors and the components of all the vectors: C = A 1 + A 2 + A A n, (5) C = A 1 + A 2 + A A n. Note: the sums here are algebraic, so please mind the ± signs of the components. 2
3 For the 3D vectors, there are similar formulae, but there is one more algebraic sum for the z components: C = A 1 + A 2 + A A n, C = A 1 + A 2 + A A n, (6) C z = A 1z + A 2z + A 3z + + A nz. Vector subtraction in components works similar to vector addition. To get B = C A (7) in components, subtract the components of A from the components of C, B = C A, B = C A, (8) in 3D also B z = C z A z. Conversion from magnitude and direction to components. A vector quantit V has magnitude and direction. On a graph the magnitude is shown b the length of the arrowed line; algebraicall, the magnitude is a non-negative number V. In a 2D plane, the direction of a vector can be specified b the angle φ v it makes with the ais, for eample V φ v (9) Now let us draw a similar diagram which also includes the components (V,V ) of the 3
4 vector V: V V (10) φ v V Note the right triangle made b the three red lines; taking the ratios of this triangle s sides and appling basic trigonometr, we immediatel obtain V V = cosφ v, V V = sinφ v, (11) and therefore V = V cosφ v, V = V sinφ v. (12) The triangle on the diagram (10) is drawn for direction of Vin the first quadrant of the coordinate sstem (between the positive and positive direction, φ v < 90 ), but the formulae (12) for the components work for all possible directions, provided we alwas measure the angle φ v counterclockwise from the positive ais. For eample, for V in the second quadrant 90 < φ v < 180, V = V cosφ v < 0, V V = V sinφ v > 0, V φ v V (13) 4
5 Likewise, for V in the third quadrant 180 < φ v < 270, V = V cosφ v < 0, V = V sinφ v < 0, φ v V V V (14) or for V in the fourth quadrant 270 < φ v < 360, V = V cosφ v > 0, V = V sinφ v < 0, φ v V V V (15) 5
6 Conversion from components to magnitude and direction. Now supposed we know the (V,V ) components of a vector V; how do we find the vector s magnitude and direction? The magnitude V follows from the Pthagoras theorem for the right triangles on an of the diagrams on the last two pages: V 2 = V 2 + V 2 (16) regardless of the signs of the V and V and therefore V = V 2 + V 2. (17) To find the direction of V, we need a bit of trigonometr. Let s take the ratio of the two equations (12) for the components (V,V ): V V = V sinφ v V cosφ v = sinφ v cosφ v = tanφ v. (18) Thus the ratio V /V c gives us the tangent of the angle φ v, so naivel we ma calculate the angle φ v itself as the arc-tangent (the inverse tangent) of this ratio, φ v?? = arctan V V. (19) However, the formula ma be off b 180, so it might give us precisel the opposite direction. Indeed, the vectors V and V have opposite directions but similar ratios This ambiguit is related to the trigonometric identit V V = V V. (20) for an angle ϕ, tan(ϕ) = tan(ϕ±180 ). (21) 6
7 Therefore, given the components of a vector, its direction is φ v = either arctan V V or arctan V V ± 180, depending on the signs of the V and V. (22) Using conversion to add vectors. Consider a simple problem: A person rides a bike for 10.0 km in the direction 30 (counterclockwise from the ais), then changes direction to 150 (also counterclockwise from the ais) and rides for 20.0 km. Find his net displacement vector. D 2 D net = D 1 + D 2 D net (23) D 1 To solve this problem, we start b converting the displacement vectors D 1 and D 2 into components: D 1 = 10.0 km cos(30 ) = km, D 1 = (10.0 km;30 ) = D 1 = 10.0 km sin(30 ) = km, (24) D 2 = 20.0 km cos(150 ) = km, D 2 = (20.0 km;150 ) = D 2 = 20.0 km sin(150 ) = km. Net, we add the two vectors in components: D net D net = D 1 + D 2 = km km = 8.66 km, = D 1 + D 2 = km km = km. (25) Finall, we convert the components of the net displacement vector into its direction and 7
8 magnitude. For the magnitude we have and hence As to the direction, D net 2 = ( D net ) 2 ( ) + D net 2 = ( 8.66 km) 2 + ( km) 2 = 75 km km 2 = 300 km 2 (26) D net = 300 km km. (27) D net D net = km 8.66 km Dnet = 1.73 = arctan D net = 60. (28) The 60 angle is the same as = +300, which correspond to the direction of D net being in the fourth quadrant. However, the signs of the components D net < 0, D net > 0 show that the direction of D net is in the second quadrant. This means that the arc-tangent is off b 180, so the correct direction of the net displacement vector is φ net = = +120 (counterclockwise from the ais). (29) The above eample illustrate a general rule for calculating sums of several vectors, A net = A 1 + A A n. (30) (1) First, convert all the vectors into components, A i, = A i cosφ i, A i, = A i sinφ i, for i = 1,2,...,n. (31) (2) Second, add the vectors in components, A net = A 1, + A 2, + + A n,, A net = A 1, + A 2, + + A n,. (32) (3) Finall, convert the components of the A net into its magnitude and direction. 8
9 Navigation Convention In navigation, the directions are given b angles counted clockwise from North instead of counterclockwise from the ais (whatever that might be). For eample, an airplane fling in the Southwest direction which is 225 clockwise from North is said to have heading 225. N 225 W E (33) v plane S To avoid confusion when ou work a navigation-related problem, it is best to avoid the and aes altogether and use the North and East aes instead of them. In particular, for the vectors ou should use the North and East components instead of the and components. In this convention, V N = V cosφ v, V E = V sinφ v, for φ v counted clockwise from North. (34) Foreample, foraplaneflingatspeedv = 220MPHintheSouthwest direction(φ v = 225 ), the velocit vector v has components v N = 220 MPH cos225 = 155 MPH, v E = 220 MPH sin225 = 155 MPH. (35) Note negative signs of the components since the plane is fling South and West instead of North and East. 9
10 Multipling vectors b scalars. The product of a scalar s and a vector V is a vector s V. Its magnitude is the absolute value of s times the magnitude of V, s V = s V. (36) The direction of the product sv is the same as direction of V for positive s but opposite from the direction of V for negative s. (For s = 0 the product sv = 0 the zero vector and its direction is undefined.) In components, B = sa has B = s A, B = s A, and in 3D also B z = s A z. (37) The product of a vector and a scalar obes the usual algebraic rules for opening parentheses: s ( V1 + V ) 2 = sv1 + sv 2, s ( V1 V ) 2 = sv1 sv 2, (s 1 +s 2 ) V = s 1 V + s2 V, (s 1 s 2 ) V = s 1 V s2 V, (38) ( ) s 1 s2v = (s1 s 2 ) V,... Phsical Eample: For a motion at constant velocit vector v i.e., motion at constant speed in a constant direction the displacement vector after time t is given b D = t v. (39) Interms ofthe time-dependent positionvector (AKAradius-vector) R(t) ofthemoving bod whose components R (t) and R (t) are simpl the time-dependent coordinates (,) of 10
11 the bod the displacement vector is the vector difference D = R = R(t) R 0, (40) so eq. (39) becomes R(t) = R 0 + t v. (41) In components, this formula means (t) = 0 + t v and (t) = 0 + t v (42) uniform motion in both and directions. Division: You cannot divide a scalar b vector, or a vector b another vector. However, ou can divide a vector b a scalar simpl multipl the vector b the inverse scalar, V s def = 1 s V. (43) For eample, to find the average velocit vector of some bod, divide its displacement vector b the time this displacement took, v avg = R t. (44) In the limit of a ver short time interval, this formula gives ou the instantaneous velocit vector v(t) = lim t 0 R(t+ t) R(t). (45) t In general, the velocit vector changes with time, which leads to the acceleration vector v(t+ t) v(t) a(t) = lim. (46) t 0 t 11
12 Motion at constant acceleration. Consider motion of some bod having a constant acceleration vector a. (Neither direction nor magnitude of a changes with time.) The velocit vector of such a bod changes with time according to v(t) = v 0 + t a (47) where v 0 is the initial velocit vector at time t 0 = 0. Note: eq. (46) looks simple in vector notations, or in components v (t) = v 0 + t a, v (t) = v 0 + t a, (48) but it leads to rather complicated formulae for the time dependence of the speed v(t) and the direction of motion. The time-dependent position vector R(t) of a uniforml accelerating bod is given b R(t) = R 0 + t v t2 a, (49) or in components (t) = 0 + t v t2 a, (t) = 0 + t v t2 a. (50) Projectile Motion A projectile is an object ou shoot, kick, throw, or otherwise send fling towards a target, for eample a basketball, a bullet, or a grenade. In phsics, projectile motion is a motion of a bod that has been released with some initial velocit (which generall has both horizontal and vertical components) and then flies free from forces other than gravit and air resistance. When the air resistance ma be neglected which is the onl case we shall stud in this class the projectile has a constant acceleration vector a = g due to gravit. Consequentl, the projectile s velocit vector v(t) and position vector R(t) evolve with time according to eqs. (47) through (50). 12
13 A projectile moves in a vertical plane, so its motion can be described using 2D vectors and their components. Let s use a coordinate sstem where the ais points verticall up while the ais is horizontal. (In 3D, the ais points along the horizontal components of the initial velocit.) In these coordinates, the downward acceleration vector a = g has components a = 0, a = g. (51) Consequentl, eqs. (48) and (50) become v (t) = v 0 = const (time independent), (t) = 0 + v 0 t, v (t) = v 0 g t, (52) (t) = 0 + v 0 t g 2 t2. Note that the first two of these equations which describe the horizontal motion are completel independent from the last two equations describing the vertical motion. Thus, the projectile moves horizontall at constant velocit as if there were no vertical motion, and at the same time it moves verticall up and down at constant acceleration as if there were no horizontal motion! 13
Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second
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 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 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 informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationVector Fields and Line Integrals
Vector Fields and Line Integrals 1. Match the following vector fields on R 2 with their plots. (a) F (, ), 1. Solution. An vector, 1 points up, and the onl plot that matches this is (III). (b) F (, ) 1,.
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 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 informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
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 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 informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
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 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 informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
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 informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
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 informationPhysics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE
1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object
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 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 informationVectors. Chapter Outline. 3.1 Coordinate Systems 3.2 Vector and Scalar Quantities 3.3 Some Properties of Vectors
P U Z Z L E R When this honebee gets back to its hive, it will tell the other bees how to return to the food it has found. moving in a special, ver precisel defined pattern, the bee conves to other workers
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 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 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 informationAffine Transformations
A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates
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 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 informationCHALLENGE PROBLEMS: CHAPTER 10. Click here for solutions. Click here for answers.
CHALLENGE PROBLEMS CHALLENGE PROBLEMS: CHAPTER 0 A Click here for answers. S Click here for solutions. m m m FIGURE FOR PROBLEM N W F FIGURE FOR PROBLEM 5. Each edge of a cubical bo has length m. The bo
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More informationSupporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Algebra and coordinate geometry: Module 2. Coordinate geometry
1 Supporting Australian Mathematics Project 3 4 5 6 7 8 9 1 11 1 A guide for teachers Years 11 and 1 Algebra and coordinate geometr: Module Coordinate geometr Coordinate geometr A guide for teachers (Years
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 information1.3. 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 informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationVELOCITY, ACCELERATION, FORCE
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
More informationChapter 8. Lines and Planes. By the end of this chapter, you will
Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes
More informationSLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT
. Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail
More information3D Stress Components. From equilibrium principles: τ xy = τ yx, τ xz = τ zx, τ zy = τ yz. Normal Stresses. Shear Stresses
3D Stress Components From equilibrium principles:, z z, z z The most general state of stress at a point ma be represented b 6 components Normal Stresses Shear Stresses Normal stress () : the subscript
More informationSouth 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 information8-3 Dot Products and Vector Projections
8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. Dr Tay Seng Chuan
Ground Rules PC11 Fundamentals of Physics I Lectures 3 and 4 Motion in One Dimension Dr Tay Seng Chuan 1 Switch off your handphone and pager Switch off your laptop computer and keep it No talking while
More informationSystems of Linear Equations: Solving by Substitution
8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More informationMath 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 informationApplications of Trigonometry
5144_Demana_Ch06pp501-566 01/11/06 9:31 PM Page 501 CHAPTER 6 Applications of Trigonometr 6.1 Vectors in the Plane 6. Dot Product of Vectors 6.3 Parametric Equations and Motion 6.4 Polar Coordinates 6.5
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 information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
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 informationChapter 5: Applying Newton s Laws
Chapter 5: Appling Newton s Laws Newton s 1 st Law he 1 st law defines what the natural states of motion: rest and constant velocit. Natural states of motion are and those states are when a = 0. In essence,
More informationLecture 1 Introduction 1. 1.1 Rectangular Coordinate Systems... 1. 1.2 Vectors... 3. Lecture 2 Length, Dot Product, Cross Product 5. 2.1 Length...
CONTENTS i Contents Lecture Introduction. Rectangular Coordinate Sstems..................... Vectors.................................. 3 Lecture Length, Dot Product, Cross Product 5. Length...................................
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent
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 informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b
More informationSlope-Intercept Form and Point-Slope Form
Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.
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 informationChapter 3 & 8.1-8.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x - 4y = 10 16x + 8y = 10
Chapter 3 & 8.1-8.3 These are meant for practice. The actual test is different. Determine whether the pair of equations represents parallel lines. 1) 9 + 3 = 12 27 + 9 = 39 1) Determine whether the pair
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 informationIntroduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11
More informationTrigonometry for AC circuits
Trigonometry for AC circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationKinematic Physics for Simulation and Game Programming
Kinematic Phsics for Simulation and Game Programming Mike Baile mjb@cs.oregonstate.edu phsics-kinematic.ppt mjb October, 05 SI Phsics Units (International Sstem of Units) Quantit Units Linear position
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 informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
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 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 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 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 informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle - in particular the basic equation representing
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph
More informationLines. We have learned that the graph of a linear equation. y = mx +b
Section 0. Lines We have learne that the graph of a linear equation = m +b is a nonvertical line with slope m an -intercept (0, b). We can also look at the angle that such a line makes with the -ais. This
More informationCh 8 Potential energy and Conservation of Energy. Question: 2, 3, 8, 9 Problems: 3, 9, 15, 21, 24, 25, 31, 32, 35, 41, 43, 47, 49, 53, 55, 63
Ch 8 Potential energ and Conservation of Energ Question: 2, 3, 8, 9 Problems: 3, 9, 15, 21, 24, 25, 31, 32, 35, 41, 43, 47, 49, 53, 55, 63 Potential energ Kinetic energ energ due to motion Potential energ
More informationIn order to describe motion you need to describe the following properties.
Chapter 2 One Dimensional Kinematics How would you describe the following motion? Ex: random 1-D path speeding up and slowing down In order to describe motion you need to describe the following properties.
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
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 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 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 informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationEquation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1
Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows :- gradient = vertical horizontal horizontal A B vertical
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 informationLinear Equations in Two Variables
Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations
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 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 informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More information