Mechanical Engineer''''s Handbook pptx

The point P is the point of application of the vector, and the linepassing through P and parallel to the vector is the line of action of the vector.The point of application may be repres

Mechanical Engineer's Handbook

The series editor, J David Irwin, is one of the best-known engineering educators inthe world Irwin has been chairman of the electrical engineering department atAuburn University for 27 years.

Published books in this series:

Control of Induction Motors

Control in Robotics and Automation

1999, B K Ghosh, N Xi, and T J Tarn

Copyright # 2001 by ACADEMIC PRESS

All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida

32887-6777.

Explicit permission from Academic Press is not required to reproduce a maximum of two

®gures or tables from an Academic Press chapter in another scienti®c or research

publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given.

Academic Press

A Harcourt Science and Technology Company

525 B Street, Suite 1900, San Diego, California 92101-4495, USA

http://www.academicpress.com

Academic Press

Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK

http://www.academicpress.com

Library of Congress Catalog Card Number: 2001088196

International Standard Book Number: 0-12-471370-X

PRINTED IN THE UNITED STATES OF AMERICA

01 02 03 04 05 06 COB 9 8 7 6 5 4 3 2 1

Table of Contents

Preface xiii

Contributors xv

CHAPTER 1 Statics Dan B Marghitu, Cristian I Diaconescu, and Bogdan O Ciocirlan 1 Vector Algebra 2

1.1 Terminology and Notation 2

1.2 Equality 4

1.3 Product of a Vector and a Scalar 4

1.4 Zero Vectors 4

1.5 Unit Vectors 4

1.6 Vector Addition 5

1.7 Resolution of Vectors and Components 6

1.8 Angle between Two Vectors 7

1.9 Scalar (Dot) Product of Vectors 9

1.10 Vector (Cross) Product of Vectors 9

1.11 Scalar Triple Product of Three Vectors 11

1.12 Vector Triple Product of Three Vectors 11

1.13 Derivative of a Vector 12

2 Centroids and Surface Properties 12

2.1 Position Vector 12

2.2 First Moment 13

2.3 Centroid of a Set of Points 13

2.4 Centroid of a Curve, Surface, or Solid 15

2.5 Mass Center of a Set of Particles 16

2.6 Mass Center of a Curve, Surface, or Solid 16

2.7 First Moment of an Area 17

2.8 Theorems of Guldinus±Pappus 21

2.9 Second Moments and the Product of Area 24

2.10 Transfer Theorem or Parallel-Axis Theorems 25

2.11 Polar Moment of Area 27

2.12 Principal Axes 28

3 Moments and Couples 30

3.1 Moment of a Bound Vector about a Point 30

3.2 Moment of a Bound Vector about a Line 31

3.3 Moments of a System of Bound Vectors 32

3.4 Couples 34

v

3.5 Equivalence 35

3.6 Representing Systems by Equivalent Systems 36

4 Equilibrium 40

4.1 Equilibrium Equations 40

4.2 Supports 42

4.3 Free-Body Diagrams 44

5 Dry Friction 46

5.1 Static Coef®cient of Friction 47

5.2 Kinetic Coef®cient of Friction 47

5.3 Angles of Friction 48

References 49

CHAPTER 2 Dynamics Dan B Marghitu, Bogdan O Ciocirlan, and Cristian I Diaconescu 1 Fundamentals 52

1.1 Space and Time 52

1.2 Numbers 52

1.3 Angular Units 53

2 Kinematics of a Point 54

2.1 Position, Velocity, and Acceleration of a Point 54

2.2 Angular Motion of a Line 55

2.3 Rotating Unit Vector 56

2.4 Straight Line Motion 57

2.5 Curvilinear Motion 58

2.6 Normal and Tangential Components 59

2.7 Relative Motion 73

3 Dynamics of a Particle 74

3.1 Newton's Second Law 74

3.2 Newtonian Gravitation 75

3.3 Inertial Reference Frames 75

3.4 Cartesian Coordinates 76

3.5 Normal and Tangential Components 77

3.6 Polar and Cylindrical Coordinates 78

3.7 Principle of Work and Energy 80

3.8 Work and Power 81

3.9 Conservation of Energy 84

3.10 Conservative Forces 85

3.11 Principle of Impulse and Momentum 87

3.12 Conservation of Linear Momentum 89

3.13 Impact 90

3.14 Principle of Angular Impulse and Momentum 94

4 Planar Kinematics of a Rigid Body 95

4.1 Types of Motion 95

4.2 Rotation about a Fixed Axis 96

4.3 Relative Velocity of Two Points of the Rigid Body 97

4.4 Angular Velocity Vector of a Rigid Body 98

4.5 Instantaneous Center 100

4.6 Relative Acceleration of Two Points of the Rigid Body 102

4.7 Motion of a Point That Moves Relative to a Rigid Body 103

5 Dynamics of a Rigid Body 111

5.1 Equation of Motion for the Center of Mass 111

5.2 Angular Momentum Principle for a System of Particles 113

5.3 Equation of Motion for General Planar Motion 115

5.4 D'Alembert's Principle 117

References 117

CHAPTER 3 Mechanics of Materials Dan B Marghitu, Cristian I Diaconescu, and Bogdan O Ciocirlan 1 Stress 120

1.1 Uniformly Distributed Stresses 120

1.2 Stress Components 120

1.3 Mohr's Circle 121

1.4 Triaxial Stress 125

1.5 Elastic Strain 127

1.6 Equilibrium 128

1.7 Shear and Moment 131

1.8 Singularity Functions 132

1.9 Normal Stress in Flexure 135

1.10 Beams with Asymmetrical Sections 139

1.11 Shear Stresses in Beams 140

1.12 Shear Stresses in Rectangular Section Beams 142

1.13 Torsion 143

1.14 Contact Stresses 147

2 De¯ection and Stiffness 149

2.1 Springs 150

2.2 Spring Rates for Tension, Compression, and Torsion 150

2.3 De¯ection Analysis 152

2.4 De¯ections Analysis Using Singularity Functions 153

2.5 Impact Analysis 157

2.6 Strain Energy 160

2.7 Castigliano's Theorem 163

2.8 Compression 165

2.9 Long Columns with Central Loading 165

2.10 Intermediate-Length Columns with Central Loading 169

2.11 Columns with Eccentric Loading 170

2.12 Short Compression Members 171

3 Fatigue 173

3.1 Endurance Limit 173

3.2 Fluctuating Stresses 178

3.3 Constant Life Fatigue Diagram 178

3.4 Fatigue Life for Randomly Varying Loads 181

3.5 Criteria of Failure 183

References 187

CHAPTER 4 Theory of Mechanisms

Dan B Marghitu

1 Fundamentals 190

1.1 Motions 190

1.2 Mobility 190

1.3 Kinematic Pairs 191

1.4 Number of Degrees of Freedom 199

1.5 Planar Mechanisms 200

2 Position Analysis 202

2.1 Cartesian Method 202

2.2 Vector Loop Method 208

3 Velocity and Acceleration Analysis 211

3.1 Driver Link 212

3.2 RRR Dyad 212

3.3 RRT Dyad 214

3.4 RTR Dyad 215

3.5 TRT Dyad 216

4 Kinetostatics 223

4.1 Moment of a Force about a Point 223

4.2 Inertia Force and Inertia Moment 224

4.3 Free-Body Diagrams 227

4.4 Reaction Forces 228

4.5 Contour Method 229

References 241

CHAPTER 5 Machine Components Dan B Marghitu, Cristian I Diaconescu, and Nicolae Craciunoiu 1 Screws 244

1.1 Screw Thread 244

1.2 Power Screws 247

2 Gears 253

2.1 Introduction 253

2.2 Geometry and Nomenclature 253

2.3 Interference and Contact Ratio 258

2.4 Ordinary Gear Trains 261

2.5 Epicyclic Gear Trains 262

2.6 Differential 267

2.7 Gear Force Analysis 270

2.8 Strength of Gear Teeth 275

3 Springs 283

3.1 Introduction 283

3.2 Material for Springs 283

3.3 Helical Extension Springs 284

3.4 Helical Compression Springs 284

3.5 Torsion Springs 290

3.6 Torsion Bar Springs 292

3.7 Multileaf Springs 293

3.8 Belleville Springs 296

4 Rolling Bearings 297

4.1 Generalities 297

4.2 Classi®cation 298

4.3 Geometry 298

4.4 Static Loading 303

4.5 Standard Dimensions 304

4.6 Bearing Selection 308

5 Lubrication and Sliding Bearings 318

5.1 Viscosity 318

5.2 Petroff's Equation 323

5.3 Hydrodynamic Lubrication Theory 326

5.4 Design Charts 328

References 336

CHAPTER 6 Theory of Vibration Dan B Marghitu, P K Raju, and Dumitru Mazilu 1 Introduction 340

2 Linear Systems with One Degree of Freedom 341

2.1 Equation of Motion 342

2.2 Free Undamped Vibrations 343

2.3 Free Damped Vibrations 345

2.4 Forced Undamped Vibrations 352

2.5 Forced Damped Vibrations 359

2.6 Mechanical Impedance 369

2.7 Vibration Isolation: Transmissibility 370

2.8 Energetic Aspect of Vibration with One DOF 374

2.9 Critical Speed of Rotating Shafts 380

3 Linear Systems with Finite Numbers of Degrees of Freedom 385

3.1 Mechanical Models 386

3.2 Mathematical Models 392

3.3 System Model 404

3.4 Analysis of System Model 405

3.5 Approximative Methods for Natural Frequencies 407

4 Machine-Tool Vibrations 416

4.1 The Machine Tool as a System 416

4.2 Actuator Subsystems 418

4.3 The Elastic Subsystem of a Machine Tool 419

4.4 Elastic System of Machine-Tool Structure 435

4.5 Subsystem of the Friction Process 437

4.6 Subsystem of Cutting Process 440

References 444

CHAPTER 7 Principles of Heat Transfer Alexandru Morega 1 Heat Transfer Thermodynamics 446

1.1 Physical Mechanisms of Heat Transfer: Conduction, Convection, and Radiation 451

1.2 Technical Problems of Heat Transfer 455

2 Conduction Heat Transfer 456

2.1 The Heat Diffusion Equation 457

2.2 Thermal Conductivity 459

2.3 Initial, Boundary, and Interface Conditions 461

2.4 Thermal Resistance 463

2.5 Steady Conduction Heat Transfer 464

2.6 Heat Transfer from Extended Surfaces (Fins) 468

2.7 Unsteady Conduction Heat Transfer 472

3 Convection Heat Transfer 488

3.1 External Forced Convection 488

3.2 Internal Forced Convection 520

3.3 External Natural Convection 535

3.4 Internal Natural Convection 549

References 555

CHAPTER 8 Fluid Dynamics Nicolae Craciunoiu and Bogdan O Ciocirlan 1 Fluids Fundamentals 560

1.1 De®nitions 560

1.2 Systems of Units 560

1.3 Speci®c Weight 560

1.4 Viscosity 561

1.5 Vapor Pressure 562

1.6 Surface Tension 562

1.7 Capillarity 562

1.8 Bulk Modulus of Elasticity 562

1.9 Statics 563

1.10 Hydrostatic Forces on Surfaces 564

1.11 Buoyancy and Flotation 565

1.12 Dimensional Analysis and Hydraulic Similitude 565

1.13 Fundamentals of Fluid Flow 568

2 Hydraulics 572

2.1 Absolute and Gage Pressure 572

2.2 Bernoulli's Theorem 573

2.3 Hydraulic Cylinders 575

2.4 Pressure Intensi®ers 578

2.5 Pressure Gages 579

2.6 Pressure Controls 580

2.7 Flow-Limiting Controls 592

2.8 Hydraulic Pumps 595

2.9 Hydraulic Motors 598

2.10 Accumulators 601

2.11 Accumulator Sizing 603

2.12 Fluid Power Transmitted 604

2.13 Piston Acceleration and Deceleration 604

2.14 Standard Hydraulic Symbols 605

2.15 Filters 606

2.16 Representative Hydraulic System 607

References 610

CHAPTER 9 Control Mircea Ivanescu 1 Introduction 612

1.1 A Classic Example 613

2 Signals 614

3 Transfer Functions 616

3.1 Transfer Functions for Standard Elements 616

3.2 Transfer Functions for Classic Systems 617

4 Connection of Elements 618

5 Poles and Zeros 620

6 Steady-State Error 623

6.1 Input Variation Steady-State Error 623

6.2 Disturbance Signal Steady-State Error 624

7 Time-Domain Performance 628

8 Frequency-Domain Performances 631

8.1 The Polar Plot Representation 632

8.2 The Logarithmic Plot Representation 633

8.3 Bandwidth 637

9 Stability of Linear Feedback Systems 639

9.1 The Routh±Hurwitz Criterion 640

9.2 The Nyquist Criterion 641

9.3 Stability by Bode Diagrams 648

10 Design of Closed-Loop Control Systems by Pole-Zero Methods 649

10.1 Standard Controllers 650

10.2 P-Controller Performance 651

10.3 Effects of the Supplementary Zero 656

10.4 Effects of the Supplementary Pole 660

10.5 Effects of Supplementary Poles and Zeros 661

10.6 Design Example: Closed-Loop Control of a Robotic Arm 664

11 Design of Closed-Loop Control Systems by Frequential Methods 669

12 State Variable Models 672

13 Nonlinear Systems 678

13.1 Nonlinear Models: Examples 678

13.2 Phase Plane Analysis 681

13.3 Stability of Nonlinear Systems 685

13.4 Liapunov's First Method 688

13.5 Liapunov's Second Method 689

14 Nonlinear Controllers by Feedback Linearization 691

15 Sliding Control 695

15.1 Fundamentals of Sliding Control 695

15.2 Variable Structure Systems 700

A Appendix 703

A.1 Differential Equations of Mechanical Systems 703

A.2 The Laplace Transform 707

A.3 Mapping Contours in the s-Plane 707

A.4 The Signal Flow Diagram 712

References 714

APPENDIX Differential Equations and Systems of Differential Equations Horatiu Barbulescu 1 Differential Equations 716

1.1 Ordinary Differential Equations: Introduction 716

1.2 Integrable Types of Equations 726

1.3 On the Existence, Uniqueness, Continuous Dependence on a Parameter, and Differentiability of Solutions of Differential Equations 766

1.4 Linear Differential Equations 774

2 Systems of Differential Equations 816

2.1 Fundamentals 816

2.2 Integrating a System of Differential Equations by the Method of Elimination 819

2.3 Finding Integrable Combinations 823

2.4 Systems of Linear Differential Equations 825

2.5 Systems of Linear Differential Equations with Constant Coef®cients 835

References 845

Index 847

The purpose of this handbook is to present the reader with a teachable textthat includes theory and examples Useful analytical techniques provide thestudent and the practitioner with powerful tools for mechanical design Thisbook may also serve as a reference for the designer and as a source book forthe researcher

This handbook is comprehensive, convenient, detailed, and is a guidefor the mechanical engineer It covers a broad spectrum of critical engineer-ing topics and helps the reader understand the fundamentals

This handbook contains the fundamental laws and theories of sciencebasic to mechanical engineering including controls and mathematics Itprovides readers with a basic understanding of the subject, together withsuggestions for more speci®c literature The general approach of this bookinvolves the presentation of a systematic explanation of the basic concepts ofmechanical systems

This handbook's special features include authoritative contributions,chapters on mechanical design, useful formulas, charts, tables, and illustra-tions With this handbook the reader can study and compare the availablemethods of analysis The reader can also become familiar with the methods

of solution and with their implementation

Dan B Marghitu

xiii

Mircea Ivanescu, (611) Department of Electrical Engineering, University

of Craiova, Craiova 1100, RomaniaDan B Marghitu, (1, 51, 119, 189, 243, 339) Department of MechanicalEngineering, Auburn University, Auburn, Alabama 36849

Dumitru Mazilu, (339) Department of Mechanical Engineering, AuburnUniversity, Auburn, Alabama 36849

Alexandru Morega, (445) Department of Electrical Engineering, nica'' University of Bucharest, Bucharest 6-77206, Romania

``Politeh-P K Raju, (339) Department of Mechanical Engineering, Auburn sity, Auburn, Alabama 36849

Univer-xv

1.10 Vector (Cross) Product of Vectors 9

1.11 Scalar Triple Product of Three Vectors 11

1.12 Vector Triple Product of Three Vectors 11

1.13 Derivative of a Vector 12

2 Centroids and Surface Properties 12

2.10 Transfer Theorems or Parallel-Axis Theorems 25

2.11 Polar Moment of Area 27

2.12 Principal Axes 28

3 Moments and Couples 30

1

1 Vector Algebra

1.1 Terminology and Notation

T he characteristics of a vector are the magnitude, the orientation, and

the sense The magnitude of a vector is speci®ed by a positivenumber and a unit having appropriate dimensions No unit is stated ifthe dimensions are those of a pure number The orientation of a vector isspeci®ed by the relationship between the vector and given reference linesand=or planes The sense of a vector is speci®ed by the order of two points

on a line parallel to the vector Orientation and sense together determine thedirection of a vector The line of action of a vector is a hypothetical in®nitestraight line collinear with the vector Vectors are denoted by boldface letters,for example, a, b, A, B, CD The symbol jvj represents the magnitude (ormodule, or absolute value) of the vector v The vectors are depicted by eitherstraight or curved arrows A vector represented by a straight arrow has thedirection indicated by the arrow The direction of a vector represented by acurved arrow is the same as the direction in which a right-handed screwmoves when the screw's axis is normal to the plane in which the arrow isdrawn and the screw is rotated as indicated by the arrow

Figure 1.1 shows representations of vectors Sometimes vectors arerepresented by means of a straight or curved arrow together with a measurenumber In this case the vector is regarded as having the direction indicated

by the arrow if the measure number is positive, and the opposite direction if

References 49

Figure 1.1

A bound vector is a vector associated with a particular point P in space(Fig 1.2) The point P is the point of application of the vector, and the linepassing through P and parallel to the vector is the line of action of the vector.

The point of application may be represented as the tail, Fig 1.2a, or the head

of the vector arrow, Fig 1.2b A free vector is not associated with a particularpoint P in space A transmissible vector is a vector that can be moved alongits line of action without change of meaning

To move the body in Fig 1.3 the force vector F can be applied anywherealong the line D or may be applied at speci®c points A; B; C The force vector

F is a transmissible vector because the resulting motion is the same in allcases

The force F applied at B will cause a different deformation of the bodythan the same force F applied at a different point C The points B and C are

on the body If we are interested in the deformation of the body, the force Fpositioned at C is a bound vector

Figure 1.2

Figure 1.3

The operations of vector analysis deal only with the characteristics ofvectors and apply, therefore, to both bound and free vectors.

a body on which they act

1.3 Product of a Vector and a Scalar

DEFINITION The product of a vector v and a scalar s, s v or vs, is a vector having the

1.4 Zero Vectors

DEFINITION A zero vector is a vector that does not have a de®nite direction and whose

magnitude is equal to zero The symbol used to denote a zero vector is 0 m

1.5 Unit Vectors

DEFINITION A unit vector (versor) is a vector with the magnitude equal to 1 m

Given a vector v, a unit vector u having the same direction as v is obtained

by forming the quotient of v and jvj:

u ˆjvjv :

1.6 Vector Addition

The sum of a vector v1and a vector v2: v1‡ v2or v2‡ v1is a vector whosecharacteristics are found by either graphical or analytical processes Thevectors v1and v2add according to the parallelogram law: v1‡ v2is equal tothe diagonal of a parallelogram formed by the graphical representation of thevectors (Fig 1.4a) The vectors v1‡ v2 is called the resultant of v1 and v2.The vectors can be added by moving them successively to parallel positions

so that the head of one vector connects to the tail of the next vector Theresultant is the vector whose tail connects to the tail of the ®rst vector, andwhose head connects to the head of the last vector (Fig 1.4b)

The sum v1‡ …ÿv2† is called the difference of v1 and v2 and is denoted

by v1ÿ v2 (Figs 1.4c and 1.4d)

The sum of n vectors vi, i ˆ 1; ; n,

Pn iˆ1vi or v1‡ v2‡


‡ vn;

is called the resultant of the vectors vi, i ˆ 1; ; n

Figure 1.4

The vector addition is:

1.Commutative, that is, the characteristics of the resultant are dent of the order in which the vectors are added (commutativity):

indepen-v1‡ v2ˆ v2‡ v1:

2.Associative, that is, the characteristics of the resultant are not affected

by the manner in which the vectors are grouped (associativity):

Here R is the set of real numbers

Every vector can be regarded as the sum of n vectors …n ˆ 2; 3; † ofwhich all but one can be selected arbitrarily

1.7 Resolution of Vectors and Components

Let 1, 2, 3 be any three unit vectors not parallel to the same plane

Every vector equation v ˆ 0, where v ˆ v1 1‡ v2 2‡ v3 3, is equivalent

to three scalar equations v1ˆ 0, v2ˆ 0, v3 ˆ 0

If the unit vectors 1, 2, 3 are mutually perpendicular they form acartesian reference frame For a cartesian reference frame the followingnotation is used (Fig 1.6):

1  ; 2 ; 3 k

and

The symbol ? denotes perpendicular

When a vector v is expressed in the form v ˆ vx ‡ vy ‡ vzk where , ,

k are mutually perpendicular unit vectors (cartesian reference frame ororthogonal reference frame), the magnitude of v is given by

jvj ˆqv2‡ v2‡ v2

:The vectors vx ˆ vx , vy ˆ vy , and vz ˆ vyk are the orthogonal or rectan-gular component vectors of the vector v The measures vx, vy, vz are theorthogonal or rectangular scalar components of the vector v

If v1 ˆ v1x ‡ v1y ‡ v1zk and v2ˆ v2x ‡ v2y ‡ v2zk, then the sum ofthe vectors is

v1‡ v2ˆ …v1x‡ v2x† ‡ …v1y‡ v2y† ‡ …v1z‡ v2z†v1zk:

1.8 Angle between Two Vectors

Let us consider any two vectors a and b One can move either vector parallel

to itself (leaving its sense unaltered) until their initial points (tails) coincide

The angle between a and b is the angle y in Figs 1.7a and 1.7b The anglebetween a and b is denoted by the symbols (a; b) or (b; a) Figure 1.7crepresents the case (a, b† ˆ 0, and Fig 1.7d represents the case (a, b† ˆ 180.The direction of a vector v ˆ vx ‡ vy ‡ vzk and relative to a cartesianreference, , , k, is given by the cosines of the angles formed by the vector

and the representative unit vectors These are called direction cosines andare denoted as (Fig 1.8)

cos…v; † ˆ cos a ˆ l; cos…v; † ˆ cos b ˆ m; cos…v; k† ˆ cos g ˆ n:The following relations exist:

vx ˆ jvj cos a; vyˆ jvj cos b; vz ˆ jvj cos g:

Figure 1.7

Figure 1.8

1.9 Scalar (Dot) Product of Vectors

DEFINITION The scalar (dot) product of a vector a and a vector b is

a
b ˆ b
a ˆ jajjbj cos…a; b†:

For any two vectors a and b and any scalar s

…sa†
b ˆ s…a
b† ˆ a
…sb† ˆ sa
b mIf

a ˆ ax ‡ ay ‡ azkand

b ˆ bx ‡ by ‡ bzk;

where , , k are mutually perpendicular unit vectors, then

a
b ˆ axbx‡ ayby‡ azbz:The following relationships exist:

v ˆ vy and k
v ˆ vz:

1.10 Vector (Cross) Product of Vectors

DEFINITION The vector (cross) product of a vector a and a vector b is the vector (Fig 1.9)

where n is a unit vector whose direction is the same as the direction ofadvance of a right-handed screw rotated from a toward b, through the angle(a, b), when the axis of the screw is perpendicular to both a and b mThe magnitude of a  b is given by

ja  bj ˆ jajjbj sin…a; b†:

If a is parallel to b, ajjb, then a  b ˆ 0 The symbol k denotes parallel Therelation a  b ˆ 0 implies only that the product jajjbj sin…a; b† is equal tozero, and this is the case whenever jaj ˆ 0, or jbj ˆ 0, or sin…a; b† ˆ 0 Forany two vectors a and b and any real scalar s,

…sa†  b ˆ s…a  b† ˆ a  …sb† ˆ sa  b:

The sense of the unit vector n that appears in the de®nition of a  b depends

on the order of the factors a and b in such a way that

A set of mutually perpendicular unit vectors ; ;k is called right-handed

if  ˆ k A set of mutually perpendicular unit vectors ; ;k is called handed if  ˆ ÿk

where ; ;k are mutually perpendicular unit vectors, then a  b can beexpressed in the following determinant form:

a  b ˆ ax ay akz

bx by bz