The Cross Product of Two Vectors in Space DEFINITION. Cross Product. u * v = s ƒ u ƒƒv ƒ sin ud n
|
|
- Garey Anderson
- 6 years ago
- Views:
Transcription
1 12.4 The Cross Prodct The Cross Prodct In stdying lines in the plane, when we needed to describe how a line was tilting, we sed the notions of slope and angle of inclination. In space, we want a way to describe how a plane is tilting. We accomplish this by mltiplying two ectors in the plane together to get a third ector perpendiclar to the plane. The direction of this third ector tells s the inclination of the plane. The prodct we se to mltiply the ectors together is the ector or cross prodct, the second of the two ector mltiplication methods we stdy in calcls. Cross prodcts are widely sed to describe the effects of forces in stdies of electricity, magnetism, flid flows, and orbital mechanics. This section presents the mathematical properties that accont for the se of cross prodcts in these fields. n The Cross Prodct of Two Vectors in Space We start with two nonzero ectors and in space. If and are not parallel, they determine a plane. We select a nit ector n perpendiclar to the plane by the right-hand rle. This means that we choose n to be the nit (normal) ector that points the way yor right thmb points when yor fingers crl throgh the angle from to (Figre 12.27). Then the cross prodct * ( cross ) is the ector defined as follows. DEFINITION Cross Prodct FIGURE *. The constrction of * = s ƒ ƒƒ ƒ sin d n Unlike the dot prodct, the cross prodct is a ector. For this reason it s also called the ector prodct of and, and applies only to ectors in space. The ector * is orthogonal to both and becase it is a scalar mltiple of n. Since the sines of 0 and p are both zero, it makes sense to define the cross prodct of two parallel nonzero ectors to be 0. If one or both of and are zero, we also define * to be zero. This way, the cross prodct of two ectors and is zero if and only if and are parallel or one or both of them are zero. Parallel Vectors Nonzero ectors and are parallel if and only if * = 0. The cross prodct obeys the following laws.
2 874 Chapter 12: Vectors and the Geometry of Space n Properties of the Cross Prodct If,, and w are any ectors and r, s are scalars, then 1. srd * ssd = srsds * d 2. * s + wd = * + * w 3. s + wd * = * + w * 4. * = -s * d 5. 0 * = 0 FIGURE The constrction of *. z k i j (j i) i j j k i (i k) i y k x i j k (k j) To isalize Property 4, for example, notice that when the fingers of a right hand crl throgh the angle from to, the thmb points the opposite way and the nit ector we choose in forming * is the negatie of the one we choose in forming * (Figre 12.28). Property 1 can be erified by applying the definition of cross prodct to both sides of the eqation and comparing the reslts. Property 2 is proed in Appendix 6. Property 3 follows by mltiplying both sides of the eqation in Property 2 by -1 and reersing the order of the prodcts sing Property 4. Property 5 is a definition. As a rle, cross prodct mltiplication is not associatie so s * d * w does not generally eqal * s * wd. (See Additional Exercise 15.) When we apply the definition to calclate the pairwise cross prodcts of i, j, and k, we find (Figre 12.29) k j i * j = -sj * id = k j * k = -sk * jd = i k * i = -si * kd = j i Diagram for recalling these prodcts FIGURE The pairwise cross prodcts of i, j, and k. and i * i = j * j = k * k = 0. ƒ * ƒ Is the Area of a Parallelogram Becase n is a nit ector, the magnitde of * is Area base height sin h sin FIGURE The parallelogram determined by and. ƒ * ƒ = ƒ ƒƒƒ ƒ sin ƒƒnƒ = ƒ ƒƒƒ sin. This is the area of the parallelogram determined by and (Figre 12.30), ƒ ƒ being the base of the parallelogram and ƒ ƒƒsin ƒ the height. Determinant Formla for * Or next objectie is to calclate * from the components of and relatie to a Cartesian coordinate system.
3 ` 12.4 The Cross Prodct 875 Determinants 2 * 2 and 3 * 3 determinants are ealated as follows: EXAMPLE ` a 1 a 2 a 3 3 b 1 b 2 b 3 3 c 1 c 2 c 3 b 1 b 3 b 1 b 2 - a 2 ` ` + a 3 ` ` c 3 c 2 c 1 a ` c EXAMPLE b = ad - bc d` = s2ds3d - s1ds -4d = = 10 b 2 b 3 = a 1 ` ` c = s -5d ` 3 1` - s3d ` s1d ` -4 1` -4 3` = -5s1-3d - 3s2 + 4d + 1s6 + 4d = = 2 (For more information, see the Web site at c 2 c 1 Sppose that = 1 i + 2 j + 3 k, = 1 i + 2 j + 3 k. Then the distribtie laws and the rles for mltiplying i, j, and k tell s that * = s 1 i + 2 j + 3 kd * s 1 i + 2 j + 3 kd The terms in the last line are the same as the terms in the expansion of the symbolic determinant We therefore hae the following rle. = 1 1 i * i i * j i * k j * i j * j j * k k * i k * j k * k = s di - s dj + s dk Calclating Cross Prodcts Using Determinants If = 1 i + 2 j + 3 k and = 1 i + 2 j + 3 k, then * = EXAMPLE 1 Calclating Cross Prodcts with Determinants z Find * and * if = 2i + j + k and = -4i + 3j + k. Soltion R( 1, 1, 2) * = = p 3 1 p i - p p j + p p k P(1, 1, 0) 0 y = -2i - 6j + 10k * = -s * d = 2i + 6j - 10k x Q(2, 1, 1) EXAMPLE 2 Finding Vectors Perpendiclar to a Plane FIGURE The area of triangle PQR is half of ƒ PQ 1 * PR 1 ƒ (Example 2). Find a ector perpendiclar to the plane of (Figre 12.31). Ps1, -1, 0d, Qs2, 1, -1d, and Rs -1, 1, 2d
4 876 Chapter 12: Vectors and the Geometry of Space Soltion The ector PQ 1 * PR 1 is perpendiclar to the plane becase it is perpendiclar to both ectors. In terms of components, PQ 1 = s2-1di + s1 + 1dj + s -1-0dk = i + 2j - k PR 1 = s -1-1di + s1 + 1dj + s2-0dk = -2i + 2j + 2k PQ 1 * PR 1 = = ` 2 2` i - ` ` j + ` ` k = 6i + 6k. EXAMPLE 3 Finding the Area of a Triangle Find the area of the triangle with ertices Ps1, -1, 0d, Qs2, 1, -1d, (Figre 12.31). and Rs -1, 1, 2d Soltion The area of the parallelogram determined by P, Q, and R is ƒ PQ 1 * PR 1 ƒ = ƒ 6i + 6k ƒ Vales from Example 2. = 2s6d 2 + s6d 2 = 22 # 36 = 622. The triangle s area is half of this, or 322. Torqe n EXAMPLE 4 Finding a Unit Normal to a Plane Find a nit ector perpendiclar to the plane of Ps1, -1, 0d, Qs2, 1, -1d, and Rs -1, 1, 2d. Soltion Since PQ 1 * PR 1 is perpendiclar to the plane, its direction n is a nit ector perpendiclar to the plane. Taking ales from Examples 2 and 3, we hae n = PQ1 * PR 1 ƒ PQ 1 * PR 1 6i + 6k = = 1 ƒ i k. For ease in calclating the cross prodct sing determinants, we sally write ectors in the form = 1 i + 2 j + 3 k rather than as ordered triples = 8 1, 2, 3 9. Component of F perpendiclar to r. Its length is F sin. r F FIGURE The torqe ector describes the tendency of the force F to drie the bolt forward. Torqe When we trn a bolt by applying a force F to a wrench (Figre 12.32), the torqe we prodce acts along the axis of the bolt to drie the bolt forward. The magnitde of the torqe depends on how far ot on the wrench the force is applied and on how mch of the force is perpendiclar to the wrench at the point of application. The nmber we se to measre the torqe s magnitde is the prodct of the length of the leer arm r and the scalar component of F perpendiclar to r. In the notation of Figre 12.32, Magnitde of torqe ector = ƒ r ƒ ƒ F ƒ sin,
5 12.4 The Cross Prodct 877 or ƒ r * F ƒ. If we let n be a nit ector along the axis of the bolt in the direction of the torqe, then a complete description of the torqe ector is r * F, or Torqe ector = s ƒ r ƒƒf ƒ sin d n. Recall that we defined * to be 0 when and are parallel. This is consistent with the torqe interpretation as well. If the force F in Figre is parallel to the wrench, meaning that we are trying to trn the bolt by pshing or plling along the line of the wrench s handle, the torqe prodced is zero. P 3 ft bar 70 F Q 20 lb magnitde force FIGURE The magnitde of the torqe exerted by F at P is abot 56.4 ft-lb (Example 5). EXAMPLE 5 Finding the Magnitde of a Torqe The magnitde of the torqe generated by force F at the piot point P in Figre is Triple Scalar or Box Prodct ƒ PQ 1 * F ƒ = ƒ PQ 1 ƒƒfƒ sin 70 L s3ds20ds0.94d L 56.4 ft-lb. The prodct s * d # w is called the triple scalar prodct of,, and w (in that order). As yo can see from the formla ƒ s * d # w ƒ = ƒ * ƒƒw ƒƒcos ƒ, the absolte ale of the prodct is the olme of the parallelepiped (parallelogram-sided box) determined by,, and w (Figre 12.34). The nmber ƒ * ƒ is the area of the base parallelogram. The nmber ƒ w ƒƒcos ƒ is the parallelepiped s height. Becase of this geometry, s * d # w is also called the box prodct of,, and w. w Height w cos Area of base Volme area of base height w cos ( ) w FIGURE The nmber is the olme of a parallelepiped. ƒ s * d # w ƒ The dot and cross may be interchanged in a triple scalar prodct withot altering its ale. By treating the planes of and w and of w and as the base planes of the parallelepiped determined by,, and w, we see that s * d # w = s * wd # = sw * d #. Since the dot prodct is commtatie, we also hae s * d # w = # s * wd.
6 878 Chapter 12: Vectors and the Geometry of Space The triple scalar prodct can be ealated as a determinant: s * d # w = c ` ` i - ` ` j + ` ` k d # w = w 1 ` ` - w 2 ` ` + w 3 ` ` = w 1 w 2 w Calclating the Triple Scalar Prodct s * d # w = w 1 w 2 w 3 EXAMPLE 6 Finding the Volme of a Parallelepiped Find the olme of the box (parallelepiped) determined by = i + 2j - k, = -2i + 3k, and w = 7j - 4k. Soltion Using the rle for calclating determinants, we find s * d # w = = The olme is ƒ s * d # w ƒ = 23 nits cbed.
Lesson 81: The Cross Product of Vectors
Lesson 8: The Cross Prodct of Vectors IBHL - SANTOWSKI In this lesson yo will learn how to find the cross prodct of two ectors how to find an orthogonal ector to a plane defined by two ectors how to find
More information6.4 VECTORS AND DOT PRODUCTS
458 Chapter 6 Additional Topics in Trigonometry 6.4 VECTORS AND DOT PRODUCTS What yo shold learn ind the dot prodct of two ectors and se the properties of the dot prodct. ind the angle between two ectors
More informationVectors in Rn un. This definition of norm is an extension of the Pythagorean Theorem. Consider the vector u = (5, 8) in R 2
MATH 307 Vectors in Rn Dr. Neal, WKU Matrices of dimension 1 n can be thoght of as coordinates, or ectors, in n- dimensional space R n. We can perform special calclations on these ectors. In particlar,
More informationLecture 3. (2) Last time: 3D space. The dot product. Dan Nichols January 30, 2018
Lectre 3 The dot prodct Dan Nichols nichols@math.mass.ed MATH 33, Spring 018 Uniersity of Massachsetts Janary 30, 018 () Last time: 3D space Right-hand rle, the three coordinate planes 3D coordinate system:
More informationLecture 9: 3.4 The Geometry of Linear Systems
Lectre 9: 3.4 The Geometry of Linear Systems Wei-Ta Ch 200/0/5 Dot Prodct Form of a Linear System Recall that a linear eqation has the form a x +a 2 x 2 + +a n x n = b (a,a 2,, a n not all zero) The corresponding
More informationSECTION 6.7. The Dot Product. Preview Exercises. 754 Chapter 6 Additional Topics in Trigonometry. 7 w u 7 2 =?. 7 v 77w7
754 Chapter 6 Additional Topics in Trigonometry 115. Yo ant to fly yor small plane de north, bt there is a 75-kilometer ind bloing from est to east. a. Find the direction angle for here yo shold head the
More informationMath 144 Activity #10 Applications of Vectors
144 p 1 Math 144 Actiity #10 Applications of Vectors In the last actiity, yo were introdced to ectors. In this actiity yo will look at some of the applications of ectors. Let the position ector = a, b
More informationCONTENTS. INTRODUCTION MEQ curriculum objectives for vectors (8% of year). page 2 What is a vector? What is a scalar? page 3, 4
CONTENTS INTRODUCTION MEQ crriclm objectives for vectors (8% of year). page 2 What is a vector? What is a scalar? page 3, 4 VECTOR CONCEPTS FROM GEOMETRIC AND ALGEBRAIC PERSPECTIVES page 1 Representation
More informationWhich of these statements are true? A) 1, 2 and 3 only C) 2, 4 and 5 only. B) 1, 2 and 5 only D) 1, 3 and 4 only
Name : 1 Qadrilateral RSTU is a parallelogram and M is the point of intersection of its diagonals. S M T ntoine lists the following vector operation statements: R U 1) ST + SR MU ) UT + UR SM 3) RS + RU
More informationu v u v v 2 v u 5, 12, v 3, 2 3. u v u 3i 4j, v 7i 2j u v u 4i 2j, v i j 6. u v u v u i 2j, v 2i j 9.
Section. Vectors and Dot Prodcts 53 Vocablary Check 1. dot prodct. 3. orthogonal. \ 5. proj PQ F PQ \ ; F PQ \ 1., 1,, 3. 5, 1, 3, 3., 1,, 3 13 9 53 1 9 13 11., 5, 1, 5. i j, i j. 3i j, 7i j 1 5 1 1 37
More information4.4 Moment of a Force About a Line
4.4 Moment of a orce bot a Line 4.4 Moment of a orce bot a Line Eample 1, page 1 of 3 1. orce is applied to the end of gearshift lever DE. Determine the moment of abot shaft. State which wa the lever will
More informationThe Cross Product. In this section, we will learn about: Cross products of vectors and their applications.
The Cross Product In this section, we will learn about: Cross products of vectors and their applications. THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is a
More informationObjective 1. Lesson 87: The Cross Product of Vectors IBHL - SANTOWSKI FINDING THE CROSS PRODUCT OF TWO VECTORS
Lesson 87: The Cross Product of Vectors IBHL - SANTOWSKI In this lesson you will learn how to find the cross product of two vectors how to find an orthogonal vector to a plane defined by two vectors how
More informationVectors. Vectors ( 向量 ) Representation of Vectors. Special Vectors. Equal vectors. Chapter 16
Vectors ( 向量 ) Chapter 16 2D Vectors A vector is a line which has both magnitde and direction. For example, in a weather report yo may hear a statement like the wind is blowing at 25 knots ( 海浬 ) in the
More informationSection 7.4: Integration of Rational Functions by Partial Fractions
Section 7.4: Integration of Rational Fnctions by Partial Fractions This is abot as complicated as it gets. The Method of Partial Fractions Ecept for a few very special cases, crrently we have no way to
More informationCorrection key. Example of an appropriate method. be the wind vector x = 120 and x = y = 160 and y = 10.
Correction key 1 D Example of an appropriate method /4 Let x, y be the wind vector (km) y 100, 150 x, y 10, 160 100 x, 150 y 10, 160 100 withot wind with wind 100 + x = 10 and x = 0 150 + y = 160 and y
More informationCS 450: COMPUTER GRAPHICS VECTORS SPRING 2016 DR. MICHAEL J. REALE
CS 45: COMPUTER GRPHICS VECTORS SPRING 216 DR. MICHEL J. RELE INTRODUCTION In graphics, we are going to represent objects and shapes in some form or other. First, thogh, we need to figre ot how to represent
More informationSetting The K Value And Polarization Mode Of The Delta Undulator
LCLS-TN-4- Setting The Vale And Polarization Mode Of The Delta Undlator Zachary Wolf, Heinz-Dieter Nhn SLAC September 4, 04 Abstract This note provides the details for setting the longitdinal positions
More informationm = Average Rate of Change (Secant Slope) Example:
Average Rate o Change Secant Slope Deinition: The average change secant slope o a nction over a particlar interval [a, b] or [a, ]. Eample: What is the average rate o change o the nction over the interval
More informationElements of Coordinate System Transformations
B Elements of Coordinate System Transformations Coordinate system transformation is a powerfl tool for solving many geometrical and kinematic problems that pertain to the design of gear ctting tools and
More information1 The space of linear transformations from R n to R m :
Math 540 Spring 20 Notes #4 Higher deriaties, Taylor s theorem The space of linear transformations from R n to R m We hae discssed linear transformations mapping R n to R m We can add sch linear transformations
More informationThe Cross Product. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan The Cross Product
The Cross Product MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Introduction Recall: the dot product of two vectors is a scalar. There is another binary operation on vectors
More informationMATH2715: Statistical Methods
MATH275: Statistical Methods Exercises III (based on lectres 5-6, work week 4, hand in lectre Mon 23 Oct) ALL qestions cont towards the continos assessment for this modle. Q. If X has a niform distribtion
More information3.3 Operations With Vectors, Linear Combinations
Operations With Vectors, Linear Combinations Performance Criteria: (d) Mltiply ectors by scalars and add ectors, algebraically Find linear combinations of ectors algebraically (e) Illstrate the parallelogram
More informationsin u 5 opp } cos u 5 adj } hyp opposite csc u 5 hyp } sec u 5 hyp } opp Using Inverse Trigonometric Functions
13 Big Idea 1 CHAPTER SUMMARY BIG IDEAS Using Trigonometric Fnctions Algebra classzone.com Electronic Fnction Library For Yor Notebook hypotense acent osite sine cosine tangent sin 5 hyp cos 5 hyp tan
More informationEE2 Mathematics : Functions of Multiple Variables
EE2 Mathematics : Fnctions of Mltiple Variables http://www2.imperial.ac.k/ nsjones These notes are not identical word-for-word with m lectres which will be gien on the blackboard. Some of these notes ma
More informationLecture Notes: Finite Element Analysis, J.E. Akin, Rice University
9. TRUSS ANALYSIS... 1 9.1 PLANAR TRUSS... 1 9. SPACE TRUSS... 11 9.3 SUMMARY... 1 9.4 EXERCISES... 15 9. Trss analysis 9.1 Planar trss: The differential eqation for the eqilibrim of an elastic bar (above)
More informationMath 116 First Midterm October 14, 2009
Math 116 First Midterm October 14, 9 Name: EXAM SOLUTIONS Instrctor: Section: 1. Do not open this exam ntil yo are told to do so.. This exam has 1 pages inclding this cover. There are 9 problems. Note
More informationElementary Linear Algebra
Elemetary Liear Algebra Ato & Rorres th Editio Lectre Set Chapter : Eclidea Vector Spaces Chapter Cotet Vectors i -Space -Space ad -Space Norm Distace i R ad Dot Prodct Orthogoality Geometry of Liear Systems
More informationSection 2.3. The Cross Product
Section.3. The Cross Product Recall that a vector can be uniquely determined by its length and direction. De nition. The cross product of two vectors u and v, denote by u v, is a vector with length j u
More information45. The Parallelogram Law states that. product of a and b is the vector a b a 2 b 3 a 3 b 2, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1. a c. a 1. b 1.
SECTION 10.4 THE CROSS PRODUCT 537 42. Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular. 43.
More informationDiscontinuous Fluctuation Distribution for Time-Dependent Problems
Discontinos Flctation Distribtion for Time-Dependent Problems Matthew Hbbard School of Compting, University of Leeds, Leeds, LS2 9JT, UK meh@comp.leeds.ac.k Introdction For some years now, the flctation
More informationPhysicsAndMathsTutor.com
. Two smooth niform spheres S and T have eqal radii. The mass of S is 0. kg and the mass of T is 0.6 kg. The spheres are moving on a smooth horizontal plane and collide obliqely. Immediately before the
More informationEXERCISES WAVE EQUATION. In Problems 1 and 2 solve the heat equation (1) subject to the given conditions. Assume a rod of length L.
.4 WAVE EQUATION 445 EXERCISES.3 In Problems and solve the heat eqation () sbject to the given conditions. Assme a rod of length.. (, t), (, t) (, ),, > >. (, t), (, t) (, ) ( ) 3. Find the temperatre
More informationBLOOM S TAXONOMY. Following Bloom s Taxonomy to Assess Students
BLOOM S TAXONOMY Topic Following Bloom s Taonomy to Assess Stdents Smmary A handot for stdents to eplain Bloom s taonomy that is sed for item writing and test constrction to test stdents to see if they
More informationConcept of Stress at a Point
Washkeic College of Engineering Section : STRONG FORMULATION Concept of Stress at a Point Consider a point ithin an arbitraril loaded deformable bod Define Normal Stress Shear Stress lim A Fn A lim A FS
More informationFEA Solution Procedure
EA Soltion Procedre (demonstrated with a -D bar element problem) EA Procedre for Static Analysis. Prepare the E model a. discretize (mesh) the strctre b. prescribe loads c. prescribe spports. Perform calclations
More informationChange of Variables. (f T) JT. f = U
Change of Variables 4-5-8 The change of ariables formla for mltiple integrals is like -sbstittion for single-ariable integrals. I ll gie the general change of ariables formla first, and consider specific
More information5. The Bernoulli Equation
5. The Bernolli Eqation [This material relates predominantly to modles ELP034, ELP035] 5. Work and Energy 5. Bernolli s Eqation 5.3 An example of the se of Bernolli s eqation 5.4 Pressre head, velocity
More informationAPPENDIX B MATRIX NOTATION. The Definition of Matrix Notation is the Definition of Matrix Multiplication B.1 INTRODUCTION
APPENDIX B MAIX NOAION he Deinition o Matrix Notation is the Deinition o Matrix Mltiplication B. INODUCION { XE "Matrix Mltiplication" }{ XE "Matrix Notation" }he se o matrix notations is not necessary
More informationThe Cross Product -(10.4)
The Cross Product -(10.4) Questions: What is the definition of the cross product of two vectors? Is it a scalar or vector? What can you say about vectors u and v (all possibilities) if the cross product
More informationChapter 5 Dot, Inner and Cross Products. 5.1 Length of a vector 5.2 Dot Product 5.3 Inner Product 5.4 Cross Product
Chapter 5 Dot Inner and Cross Prodcts 5. Length of a ector 5. Dot Prodct 5.3 Inner Prodct 5.4 Cross Prodct 5. Length and Dot Prodct in R n Length: The length of a ector ( n in R n is gien by n Notes: The
More informationOPTIMUM EXPRESSION FOR COMPUTATION OF THE GRAVITY FIELD OF A POLYHEDRAL BODY WITH LINEARLY INCREASING DENSITY 1
OPTIMUM EXPRESSION FOR COMPUTATION OF THE GRAVITY FIEL OF A POLYHERAL BOY WITH LINEARLY INCREASING ENSITY 1 V. POHÁNKA2 Abstract The formla for the comptation of the gravity field of a polyhedral body
More informationThe Cross Product. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) The Cross Product Spring /
The Cross Product Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) The Cross Product Spring 2012 1 / 15 Introduction The cross product is the second multiplication operation between vectors we will
More informationL = 2 λ 2 = λ (1) In other words, the wavelength of the wave in question equals to the string length,
PHY 309 L. Soltions for Problem set # 6. Textbook problem Q.20 at the end of chapter 5: For any standing wave on a string, the distance between neighboring nodes is λ/2, one half of the wavelength. The
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA. PRINCIPLES AND APPLICATIONS of FLUID MECHANICS UNIT 13 NQF LEVEL 3 OUTCOME 3 - HYDRODYNAMICS
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA PRINCIPLES AND APPLICATIONS of FLUID MECHANICS UNIT 3 NQF LEVEL 3 OUTCOME 3 - HYDRODYNAMICS TUTORIAL - PIPE FLOW CONTENT Be able to determine the parameters of pipeline
More information3.4-Miscellaneous Equations
.-Miscellaneos Eqations Factoring Higher Degree Polynomials: Many higher degree polynomials can be solved by factoring. Of particlar vale is the method of factoring by groping, however all types of factoring
More informationGraphs and Networks Lecture 5. PageRank. Lecturer: Daniel A. Spielman September 20, 2007
Graphs and Networks Lectre 5 PageRank Lectrer: Daniel A. Spielman September 20, 2007 5.1 Intro to PageRank PageRank, the algorithm reportedly sed by Google, assigns a nmerical rank to eery web page. More
More informationVectors. February 1, 2010
Vectors Febrary 1, 2010 Motivation Location of projector from crrent position and orientation: direction of projector distance to projector (direction, distance=magnitde) vector Examples: Force Velocity
More information6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line
CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle
More informationControl Systems
6.5 Control Systems Last Time: Introdction Motivation Corse Overview Project Math. Descriptions of Systems ~ Review Classification of Systems Linear Systems LTI Systems The notion of state and state variables
More informationu P(t) = P(x,y) r v t=0 4/4/2006 Motion ( F.Robilliard) 1
y g j P(t) P(,y) r t0 i 4/4/006 Motion ( F.Robilliard) 1 Motion: We stdy in detail three cases of motion: 1. Motion in one dimension with constant acceleration niform linear motion.. Motion in two dimensions
More information(arrows denote positive direction)
12 Chapter 12 12.1 3-dimensional Coordinate System The 3-dimensional coordinate system we use are coordinates on R 3. The coordinate is presented as a triple of numbers: (a,b,c). In the Cartesian coordinate
More informationIntegration of Basic Functions. Session 7 : 9/23 1
Integration o Basic Fnctions Session 7 : 9/3 Antiderivation Integration Deinition: Taking the antiderivative, or integral, o some nction F(), reslts in the nction () i ()F() Pt simply: i yo take the integral
More informationHomework 5 Solutions
Q Homework Soltions We know that the colmn space is the same as span{a & a ( a * } bt we want the basis Ths we need to make a & a ( a * linearly independent So in each of the following problems we row
More informationUsing the Biot Savart Law: The Field of a Straight Wire
Exective Editor: Nancy Whilton Project Manager: Katie Conley Development Editors: John Mrdzek, Matt Walker Editorial Assistant: arah Kabisch Development Manager: Cathy Mrphy Project Management Team Lead:
More informationIncompressible Viscoelastic Flow of a Generalised Oldroyed-B Fluid through Porous Medium between Two Infinite Parallel Plates in a Rotating System
International Jornal of Compter Applications (97 8887) Volme 79 No., October Incompressible Viscoelastic Flow of a Generalised Oldroed-B Flid throgh Poros Medim between Two Infinite Parallel Plates in
More informationEuclidean Vector Spaces
CHAPTER 3 Eclidean Vector Spaces CHAPTER CONTENTS 3.1 Vectors in 2-Space, 3-Space, and n-space 131 3.2 Norm, Dot Prodct, and Distance in R n 142 3.3 Orthogonalit 155 3.4 The Geometr of Linear Sstems 164
More informationChange of Variables. f(x, y) da = (1) If the transformation T hasn t already been given, come up with the transformation to use.
MATH 2Q Spring 26 Daid Nichols Change of Variables Change of ariables in mltiple integrals is complicated, bt it can be broken down into steps as follows. The starting point is a doble integral in & y.
More informationENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 2 LINEAR IMPULSE AND MOMENTUM
ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D5 TUTORIAL LINEAR IMPULSE AND MOMENTUM On copletion of this ttorial yo shold be able to do the following. State Newton s laws of otion. Define linear
More informationMicroscopic Properties of Gases
icroscopic Properties of Gases So far we he seen the gas laws. These came from observations. In this section we want to look at a theory that explains the gas laws: The kinetic theory of gases or The kinetic
More informationQuadratic and Rational Inequalities
Chapter Qadratic Eqations and Ineqalities. Gidelines for solving word problems: (a) Write a verbal model that will describe what yo need to know. (b) Assign labels to each part of the verbal model nmbers
More informationIf the pull is downward (Fig. 1), we want C to point into the page. If the pull is upward (Fig. 2), we want C to point out of the page.
11.5 Cross Product Contemporary Calculus 1 11.5 CROSS PRODUCT This section is the final one about the arithmetic of vectors, and it introduces a second type of vector vector multiplication called the cross
More informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
PK K I N E M A T I C S Syllabs : Frame of reference. Motion in a straight line : Position-time graph, speed and velocity. Uniform and non-niform motion, average speed and instantaneos velocity. Uniformly
More informationMAC Module 5 Vectors in 2-Space and 3-Space II
MAC 2103 Module 5 Vectors in 2-Space and 3-Space II 1 Learning Objectives Upon completing this module, you should be able to: 1. Determine the cross product of a vector in R 3. 2. Determine a scalar triple
More informationLinear System Theory (Fall 2011): Homework 1. Solutions
Linear System Theory (Fall 20): Homework Soltions De Sep. 29, 20 Exercise (C.T. Chen: Ex.3-8). Consider a linear system with inpt and otpt y. Three experiments are performed on this system sing the inpts
More information58. The Triangle Inequality for vectors is. dot product.] 59. The Parallelogram Law states that
786 CAPTER 12 VECTORS AND TE GEOETRY OF SPACE 0, 0, 1, and 1, 1, 1 as shown in the figure. Then the centroid is. ( 1 2, 1 2, 1 2 ) ] x z C 54. If c a a, where a,, and c are all nonzero vectors, show that
More information12.1 Three Dimensional Coordinate Systems (Review) Equation of a sphere
12.2 Vectors 12.1 Three Dimensional Coordinate Systems (Reiew) Equation of a sphere x a 2 + y b 2 + (z c) 2 = r 2 Center (a,b,c) radius r 12.2 Vectors Quantities like displacement, elocity, and force inole
More informationEfficiency Increase and Input Power Decrease of Converted Prototype Pump Performance
International Jornal of Flid Machinery and Systems DOI: http://dx.doi.org/10.593/ijfms.016.9.3.05 Vol. 9, No. 3, Jly-September 016 ISSN (Online): 188-9554 Original Paper Efficiency Increase and Inpt Power
More informationDiffraction of light due to ultrasonic wave propagation in liquids
Diffraction of light de to ltrasonic wave propagation in liqids Introdction: Acostic waves in liqids case density changes with spacing determined by the freqency and the speed of the sond wave. For ltrasonic
More informationThe Real Stabilizability Radius of the Multi-Link Inverted Pendulum
Proceedings of the 26 American Control Conference Minneapolis, Minnesota, USA, Jne 14-16, 26 WeC123 The Real Stabilizability Radis of the Mlti-Link Inerted Pendlm Simon Lam and Edward J Daison Abstract
More informationSection 13.4 The Cross Product
Section 13.4 The Cross Product Multiplying Vectors 2 In this section we consider the more technical multiplication which can be defined on vectors in 3-space (but not vectors in 2-space). 1. Basic Definitions
More informationLinear Strain Triangle and other types of 2D elements. By S. Ziaei Rad
Linear Strain Triangle and other tpes o D elements B S. Ziaei Rad Linear Strain Triangle (LST or T6 This element is also called qadratic trianglar element. Qadratic Trianglar Element Linear Strain Triangle
More informationSources of Non Stationarity in the Semivariogram
Sorces of Non Stationarity in the Semivariogram Migel A. Cba and Oy Leangthong Traditional ncertainty characterization techniqes sch as Simple Kriging or Seqential Gassian Simlation rely on stationary
More informationLogarithmic, Exponential and Other Transcendental Functions
Logarithmic, Eponential an Other Transcenental Fnctions 5: The Natral Logarithmic Fnction: Differentiation The Definition First, yo mst know the real efinition of the natral logarithm: ln= t (where > 0)
More informationConceptual Questions. Problems. 852 CHAPTER 29 Magnetic Fields
852 CHAPTER 29 Magnetic Fields magnitde crrent, and the niform magnetic field points in the positive direction. Rank the loops by the magnitde of the torqe eerted on them by the field from largest to smallest.
More informationVectors. chapter. Figure 3.1 Designation of points in a Cartesian coordinate system. Every point is labeled with coordinates (x, y).
chapter 3 Vectors 3.1 Coordinate stems 3.2 Vector and calar Qantities 3.3 ome Properties of Vectors 3.4 Components of a Vector and Unit Vectors In or std of phsics, we often need to work with phsical qantities
More informationGarret Sobczyk s 2x2 Matrix Derivation
Garret Sobczyk s x Matrix Derivation Krt Nalty May, 05 Abstract Using matrices to represent geometric algebras is known, bt not necessarily the best practice. While I have sed small compter programs to
More informationSareban: Evaluation of Three Common Algorithms for Structure Active Control
Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, 1638-1646 1638 Evalation of Three Common Algorithms for Strctre Active Control Mohammad Sareban Department of Civil Engineering Shahrood
More informationUNCERTAINTY FOCUSED STRENGTH ANALYSIS MODEL
8th International DAAAM Baltic Conference "INDUSTRIAL ENGINEERING - 19-1 April 01, Tallinn, Estonia UNCERTAINTY FOCUSED STRENGTH ANALYSIS MODEL Põdra, P. & Laaneots, R. Abstract: Strength analysis is a
More informationEXPT. 5 DETERMINATION OF pk a OF AN INDICATOR USING SPECTROPHOTOMETRY
EXPT. 5 DETERMITIO OF pk a OF IDICTOR USIG SPECTROPHOTOMETRY Strctre 5.1 Introdction Objectives 5.2 Principle 5.3 Spectrophotometric Determination of pka Vale of Indicator 5.4 Reqirements 5.5 Soltions
More informationCHAPTER 3 : VECTORS. Definition 3.1 A vector is a quantity that has both magnitude and direction.
EQT 101-Engineering Mathematics I Teaching Module CHAPTER 3 : VECTORS 3.1 Introduction Definition 3.1 A ector is a quantity that has both magnitude and direction. A ector is often represented by an arrow
More information1. INTRODUCTION. A solution for the dark matter mystery based on Euclidean relativity. Frédéric LASSIAILLE 2009 Page 1 14/05/2010. Frédéric LASSIAILLE
Frédéric LASSIAILLE 2009 Page 1 14/05/2010 Frédéric LASSIAILLE email: lmimi2003@hotmail.com http://lmi.chez-alice.fr/anglais A soltion for the dark matter mystery based on Eclidean relativity The stdy
More informationElectric and Magnetic Fields
Electric and Magnetic Fields Jst as we want to define an electric field for electrostatic forces, we also want to define a magnetic field for magnetic forces. E (nits of N/C, or V/m) B (nits of Tesla)
More informationThe Cross Product of Two Vectors
The Cross roduct of Two Vectors In proving some statements involving surface integrals, there will be a need to approximate areas of segments of the surface by areas of parallelograms. Therefore it is
More informationThe Dot Product Pg. 377 # 6ace, 7bdf, 9, 11, 14 Pg. 385 # 2, 3, 4, 6bd, 7, 9b, 10, 14 Sept. 25
UNIT 2 - APPLICATIONS OF VECTORS Date Lesson TOPIC Homework Sept. 19 2.1 (11) 7.1 Vectors as Forces Pg. 362 # 2, 5a, 6, 8, 10 13, 16, 17 Sept. 21 2.2 (12) 7.2 Velocity as Vectors Pg. 369 # 2,3, 4, 6, 7,
More informationMEG 741 Energy and Variational Methods in Mechanics I
MEG 74 Energ and Variational Methods in Mechanics I Brendan J. O Toole, Ph.D. Associate Professor of Mechanical Engineering Howard R. Hghes College of Engineering Universit of Nevada Las Vegas TBE B- (7)
More informationTechnical Note. ODiSI-B Sensor Strain Gage Factor Uncertainty
Technical Note EN-FY160 Revision November 30, 016 ODiSI-B Sensor Strain Gage Factor Uncertainty Abstract Lna has pdated or strain sensor calibration tool to spport NIST-traceable measrements, to compte
More informationThis Topic follows on from Calculus Topics C1 - C3 to give further rules and applications of differentiation.
CALCULUS C Topic Overview C FURTHER DIFFERENTIATION This Topic follows on from Calcls Topics C - C to give frther rles applications of differentiation. Yo shold be familiar with Logarithms (Algebra Topic
More information10.2 Solving Quadratic Equations by Completing the Square
. Solving Qadratic Eqations b Completing the Sqare Consider the eqation ( ) We can see clearl that the soltions are However, What if the eqation was given to s in standard form, that is 6 How wold we go
More informationFRTN10 Exercise 12. Synthesis by Convex Optimization
FRTN Exercise 2. 2. We want to design a controller C for the stable SISO process P as shown in Figre 2. sing the Yola parametrization and convex optimization. To do this, the control loop mst first be
More informationPrandl established a universal velocity profile for flow parallel to the bed given by
EM 0--00 (Part VI) (g) The nderlayers shold be at least three thicknesses of the W 50 stone, bt never less than 0.3 m (Ahrens 98b). The thickness can be calclated sing Eqation VI-5-9 with a coefficient
More informationMomentum Equation. Necessary because body is not made up of a fixed assembly of particles Its volume is the same however Imaginary
Momentm Eqation Interest in the momentm eqation: Qantification of proplsion rates esign strctres for power generation esign of pipeline systems to withstand forces at bends and other places where the flow
More informationPhysicsAndMathsTutor.com
C Integration - By sbstittion PhysicsAndMathsTtor.com. Using the sbstittion cos +, or otherwise, show that e cos + sin d e(e ) (Total marks). (a) Using the sbstittion cos, or otherwise, find the eact vale
More informationThe Theory of Virtual Particles as an Alternative to Special Relativity
International Jornal of Physics, 017, Vol. 5, No. 4, 141-146 Available online at http://pbs.sciepb.com/ijp/5/4/6 Science and Edcation Pblishing DOI:10.1691/ijp-5-4-6 The Theory of Virtal Particles as an
More informationSolutions to Math 152 Review Problems for Exam 1
Soltions to Math 5 Review Problems for Eam () If A() is the area of the rectangle formed when the solid is sliced at perpendiclar to the -ais, then A() = ( ), becase the height of the rectangle is and
More informationHomogeneous Liner Systems with Constant Coefficients
Homogeneos Liner Systems with Constant Coefficients Jly, 06 The object of stdy in this section is where A is a d d constant matrix whose entries are real nmbers. As before, we will look to the exponential
More informationThe Dual of the Maximum Likelihood Method
Department of Agricltral and Resorce Economics University of California, Davis The Dal of the Maximm Likelihood Method by Qirino Paris Working Paper No. 12-002 2012 Copyright @ 2012 by Qirino Paris All
More informationReduction of over-determined systems of differential equations
Redction of oer-determined systems of differential eqations Maim Zaytse 1) 1, ) and Vyachesla Akkerman 1) Nclear Safety Institte, Rssian Academy of Sciences, Moscow, 115191 Rssia ) Department of Mechanical
More information2-9. The plate is subjected to the forces acting on members A and B as shown. If θ = 60 o, determine the magnitude of the resultant of these forces
2-9. The plate is subjected to the forces acting on members A and B as shown. If θ 60 o, determine the magnitude of the resultant of these forces and its direction measured clockwise from the positie x
More information