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


 Andra Warren
 5 years ago
 Views:
Transcription
1 Vector has a magnitude and a direction Scalar has a magnitude
2 Vector has a magnitude and a direction Scalar has a magnitude a brick on a table
3 Vector has a magnitude and a direction Scalar has a magnitude brick moves (2 meters)
4 Vector has a magnitude and a direction Scalar has a magnitude brick moves (2 meters) Brick moved 2 meters.
5 Vector has a magnitude and a direction Scalar has a magnitude brick moves (2 meters) Brick moved 2 meters to the right. Brick moved 2 meters.
6 Vector has a magnitude and a direction Scalar has a magnitude brick moves (2 meters) Brick moved 2 meters to the right. direction specified Brick moved 2 meters.
7 Vector has a magnitude and a direction Scalar has a magnitude brick moves (2 meters) Brick moved 2 meters to the right. s = 2 m to the right displacement direction specified Brick moved 2 meters. s = 2 m distance
8 Vector has a magnitude and a direction Scalar has a magnitude brick moves (2 meters) Brick moved 2 meters to the right. s = 2 m to the right direction specified Brick moved 2 meters. s = 2 m a vector displacement a scalar distance
9 Some quantities in physics Vector displacement s or s Scalar distance s different notations
10 Some quantities in physics Vector displacement s or s velocity v or v Scalar distance s speed, velocity v
11 Some quantities in physics Vector displacement s or s velocity v or v acceleration a or a Scalar distance s speed, velocity v acceleration a
12 Some quantities in physics Vector displacement s or s velocity v or v acceleration a or a force F or F Scalar distance s speed, velocity v acceleration a force F
13 Some quantities in physics Vector displacement s or s velocity v or v acceleration a or a force F or F Scalar distance s speed, velocity v acceleration a force F time t mass m
14 Working in 3D The moving brick was a 1D situation brick moves (2 meters) We will work primarily in 3D in this class.
15 Working in 3D The moving brick was a 1D situation brick moves (2 meters) We will work primarily in 3D in this class. Set up a righthanded coordinate system: y y z x x z righthanded system lefthanded system
16 Working in 3D The moving brick was a 1D situation brick moves (2 meters) We will work primarily in 3D in this class. Set up a righthanded coordinate system: y y x The convention is to use a righthanded system. z x z righthanded system lefthanded system
17 Displacement in 3D Something moves 3 m in the x direction, and 2 m in the y direction, and 6 m in the z direction
18 Displacement in 3D Something moves 3 m in the x direction, and 2 m in the y direction, and 6 m in the z direction = s a displacement in 3D
19 Displacement in 3D Something moves 3 m in the x direction, and 2 m in the y direction, and 6 m in the z direction = s a displacement in 3D Much better notation s = (3 m, 2 m, 6 m)
20 Displacement in 3D Something moves 3 m in the x direction, and 2 m in the y direction, and 6 m in the z direction = s a displacement in 3D Much better notation s = (3 m, 2 m, 6 m) More generally s = (s x, s y, s z ) components of the vector s
21 Visualizing the components s = (s x, s y, s z ) y s x z
22 Visualizing the components s = (s x, s y, s z ) s y y s z s s x (cf. actual physical model of this set up in class) x z
23 Visualizing the components s = (s x, s y, s z ) s y y s z s s x (cf. actual physical model of this set up in class) x z Magnitude of the vector? Pythagorean theorem gives it to us s = s = s2 x + s2 y + s2 z
24 A scalar times a vector Multiplying a vector by a scalar just scales the length of the vector. s = (s x, s y, s z ) as = (as x, as y, as z )
25 A scalar times a vector Multiplying a vector by a scalar just scales the length of the vector. s = (s x, s y, s z ) as = (as x, as y, as z ) y s 2s x z
26 Adding vectors Just add components. obvious for something like displacement s = (s x, s y, s z ) t = (t x, t y, t z ) s + t = (s x +t x, s y +t y, s z +t z )
27 Adding vectors Just add components. obvious for something like displacement s = (s x, s y, s z ) t = (t x, t y, t z ) Graphically + = s + t = (s x +t x, s y +t y, s z +t z )
28 Unit vectors Using two preceding rules (for as and s 1 +s 2 ), we can make a new useful notation using unit vectors. Define three unit vectors: x = (1, 0, 0) y = (0, 1, 0) z = (0, 0, 1) hat says that magnitude = 1
29 Unit vectors Using two preceding rules (for as and s 1 +s 2 ), we can make a new useful notation using unit vectors. Define three unit vectors: x = (1, 0, 0) y = (0, 1, 0) z = (0, 0, 1) hat says that magnitude = 1 Can assemble any vector by multiplying and adding these three vectors together
30 Unit vectors Using two preceding rules (for as and s 1 +s 2 ), we can make a new useful notation using unit vectors. Define three unit vectors: x = (1, 0, 0) y = (0, 1, 0) z = (0, 0, 1) hat says that magnitude = 1 s = (s x, s y, s z ) Can assemble any vector by multiplying and adding these three vectors together s = s x x + s y y + s z z equivalent
31 Representation is arbitrary Or: coordinate system is arbitrary Three identical vectors r 1 r 2 r 3
32 Representation is arbitrary Or: coordinate system is arbitrary Three identical vectors r 1 r 2 r 3 r 1 = r 2 = r 3 This equality is true.
33 Representation is arbitrary Or: coordinate system is arbitrary Three identical vectors No particular representation (coordinate system) has been chosen for any of these yet. r 1 r 2 r 3 r 1 = r 2 = r 3 This equality is true.
34 Representation is arbitrary Or: coordinate system is arbitrary Three identical vectors y r 1 r 2 r 3 x r 1 = r 2 = r 3 This equality is still true.
35 Choosing a particular representation is often convenient. s = (s x, s y, s z )
36 Choosing a particular representation is often convenient. s = (s x, s y, s z ) Also, thinking of the tail of the vector as being at the origin lets you think of vectors as points in 3D space. y s x z
37 Choosing a particular representation is often convenient. s = (s x, s y, s z ) Also, thinking of the tail of the vector as being at the origin lets you think of vectors as points in 3D space. y s x z But, take care! Vectors are fundamentally just lengths and directions and aren t tied to a representation. s
38 Derivative of a vector s = s x x + s y y + s z z ds ds x + ds y + ds z dt = x + y + z dt dt dt
39 Derivative of a vector s = s x x + s y y + s z z ds ds x + ds y + ds z dt = x + y + z dt dt dt If the position of an object is: Then the velocity is: x(t) v(t) = dx dt (example on blackboard)
40 Multiplying vectors (sort of) Given these objects called vectors, we can define various useful operations with them.
41 Multiplying vectors (sort of) Given these objects called vectors, we can define various useful operations with them. Two useful operations are multiplicationlike in appearance. dot product and cross product
42 Multiplying vectors (sort of) Given these objects called vectors, we can define various useful operations with them. Two useful operations are multiplicationlike in appearance. dot product and cross product For each: First, the definition. Then, some intuition.
43 Dot product s = s x x + s y y + s z z t = t x x + t y y + t z z s t = s x t x + s y t y + s z t z A dot product yields a scalar
44 Dot product s = s x x + s y y + s z z t = t x x + t y y + t z z s t = s x t x + s y t y + s z t z A dot product yields a scalar s t = s t cos θ st Alternative form: product of the magnitudes times cosine of the angle between the vectors
45 Dot product s = s x x + s y y + s z z t = t x x + t y y + t z z s t = s x t x + s y t y + s z t z A dot product yields a scalar s t = s t cos θ st Alternative form: product of the magnitudes times cosine of the angle between the vectors Clearly: s s = s 2 and s t = t s
46 Dot product Why? Gives the product of the magnitudes, with the modification that it only counts the components that are parallel. a b
47 Dot product Why? Gives the product of the magnitudes, with the modification that it only counts the components that are parallel. a θ b
48 Dot product Why? Gives the product of the magnitudes, with the modification that it only counts the components that are parallel. a θ b a cos θ b a b = a b cos θ
49 Dot product Why? Can look at it in reverse a θ b a b cos θ a b = a b cos θ
50 Dot product Bonus Can get angle between two vectors knowing only the components without any trouble Consider: s 1 = (1, 3, 4) s 2 = (2, 6, 1) What s the angle between these vectors?
51 Dot product Bonus Can get angle between two vectors knowing only the components without any trouble Consider: s 1 = (1, 3, 4) s 2 = (2, 6, 1) What s the angle between these vectors? Equate the two expressions for dot product and solve for cos θ (1)(2) + (3)(6) + (4)(1) = [ (1) 2 + (3) 2 + (4) 2 ] [ (2) 2 + (6) 2 + (1) 2 ] cos θ
52 Cross product s = s x x + s y y + s z z t = t x x + t y y + t z z s t = (s y t z  s z t y ) x + (s z t x  s x t z ) y + (s x t y  s y t x ) z A cross product yields another vector, perpendicular to the original two.
53 Cross product s = s x x + s y y + s z z t = t x x + t y y + t z z s t = (s y t z  s z t y ) x + (s z t x  s x t z ) y + (s x t y  s y t x ) z A cross product yields another vector, perpendicular to the original two. s t = s t sin θ st u Alternative form. Here, the vector u points perpendicular to the plane of s and t, according to a righthand rule
54 Cross product s = s x x + s y y + s z z t = t x x + t y y + t z z s t = (s y t z  s z t y ) x + (s z t x  s x t z ) y + (s x t y  s y t x ) z A cross product yields another vector, perpendicular to the original two. s t = s t sin θ st u Not commutative! Alternative form. Here, the vector u points perpendicular to the plane of s and t, according to a righthand rule s t = t s
55 Cross product Why? Gives the product of the magnitudes with the modification that it neglects any parallel components. a θ b vector pointing out of page vector pointing into page a b a sin θ b a b = a b sin θ
56 Cross product Bonus If you are familiar with matrices and determinants a b = x y z a x a y a z b x b y b z
57 Cross product Bonus If you are familiar with matrices and determinants a b = x y z a x a y a z b x b y b z For the curious: The result of a cross product is subtly different from a regular vector and is sometimes called a pseudovector or axial vector. (further discussion in class)
Figure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More 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 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 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 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 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 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 informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More 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 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 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 informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More 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 informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
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 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 information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero 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 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 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 3space. This time the outcome will be a vector in 3space. Definition
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
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 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 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 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 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 informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
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 informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
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 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 informationReview of Vector Analysis in Cartesian Coordinates
R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in threedimensional space, we also examine the
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 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 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 VECTOR PRODUCT. Graham S McDonald. A Tutorial Module for learning about the vector product of two vectors. Table of contents Begin Tutorial
Vectors VECTOR PRODUCT Graham S McDonald A Tutorial Module for learning about the vector product of two vectors Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk 1. Theory 2. Exercises
More 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 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 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 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 informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More informationChapter 6. Work and Energy
Chapter 6 Work and Energy The concept of forces acting on a mass (one object) is intimately related to the concept of ENERGY production or storage. A mass accelerated to a nonzero speed carries energy
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 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 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 1D path speeding up and slowing down In order to describe motion you need to describe the following properties.
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 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 informationAll About Motion  Displacement, Velocity and Acceleration
All About Motion  Displacement, Velocity and Acceleration Program Synopsis 2008 20 minutes Teacher Notes: Ian Walter Dip App Chem; GDipEd Admin; TTTC This program explores vector and scalar quantities
More 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 information2. Spin Chemistry and the Vector Model
2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing
More 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 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 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 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 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 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 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 informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
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 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 information1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3
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 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 informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
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 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 informationChapter 4. Moment  the tendency of a force to rotate an object
Chapter 4 Moment  the tendency of a force to rotate an object Finding the moment  2D Scalar Formulation Magnitude of force Mo = F d Rotation is clockwise or counter clockwise Moment about 0 Perpendicular
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 informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More 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 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 informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationMechanics 1: Vectors
Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationGeneral Physics 1. Class Goals
General Physics 1 Class Goals Develop problem solving skills Learn the basic concepts of mechanics and learn how to apply these concepts to solve problems Build on your understanding of how the world works
More informationLecture 7. Matthew T. Mason. Mechanics of Manipulation. Lecture 7. Representing Rotation. Kinematic representation: goals, overview
Matthew T. Mason Mechanics of Manipulation Today s outline Readings, etc. We are starting chapter 3 of the text Lots of stuff online on representing rotations Murray, Li, and Sastry for matrix exponential
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 informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More 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 informationOrthogonal Projections and Orthonormal Bases
CS 3, HANDOUT A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus 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 information10.5. Click here for answers. Click here for solutions. EQUATIONS OF LINES AND PLANES. 3x 4y 6z 9 4, 2, 5. x y z. z 2. x 2. y 1.
SECTION EQUATIONS OF LINES AND PLANES 1 EQUATIONS OF LINES AND PLANES A Click here for answers. S Click here for solutions. 1 Find a vector equation and parametric equations for the line passing through
More informationMidterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m
Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of
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 informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
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 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 informationCBE 6333, R. Levicky 1 Differential Balance Equations
CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,
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 informationScalar versus Vector Quantities. Speed. Speed: Example Two. Scalar Quantities. Average Speed = distance (in meters) time (in seconds) v =
Scalar versus Vector Quantities Scalar Quantities Magnitude (size) 55 mph Speed Average Speed = distance (in meters) time (in seconds) Vector Quantities Magnitude (size) Direction 55 mph, North v = Dx
More informationThe Geometry of the Dot and Cross Products
Journal of Online Mathematics and Its Applications Volume 6. June 2006. Article ID 1156 The Geometry of the Dot and Cross Products Tevian Dray Corinne A. Manogue 1 Introduction Most students first learn
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 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 information