Technology & Engineering
Scalar & Vector Geometry
Scalar and vector geometry are fundamental concepts in mathematics and physics. Scalars are quantities that are fully described by their magnitude, such as temperature or mass, while vectors have both magnitude and direction, like force or velocity. In engineering, understanding scalar and vector geometry is crucial for analyzing and solving problems related to forces, motion, and other physical phenomena.
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9 Key excerpts on "Scalar & Vector Geometry"
eBook - PDFMathematical Methods for Physicists
A Concise Introduction
- Tai L. Chow(Author)
- 2000(Publication Date)
- Cambridge University Press(Publisher)
1 Vector and tensor analysis Vectors and scalars Vector methods have become standard tools for the physicists. In this chapter we discuss the properties of the vectors and vector fields that occur in classical physics. We will do so in a way, and in a notation, that leads to the formation of abstract linear vector spaces in Chapter 5. A physical quantity that is completely specified, in appropriate units, by a single number (called its magnitude) such as volume, mass, and temperature is called a scalar. Scalar quantities are treated as ordinary real numbers. They obey all the regular rules of algebraic addition, subtraction, multiplication, division, and so on. There are also physical quantities which require a magnitude and a direction for their complete specification. These are called vectors if their combination with each other is commutative (that is the order of addition may be changed without aecting the result). Thus not all quantities possessing magnitude and direction are vectors. Angular displacement, for example, may be characterised by magni-tude and direction but is not a vector, for the addition of two or more angular displacements is not, in general, commutative (Fig. 1.1). In print, we shall denote vectors by boldface letters (such as A ) and use ordin-ary italic letters (such as A ) for their magnitudes; in writing, vectors are usually represented by a letter with an arrow above it such as ~ A . A given vector A (or ~ A ) can be written as A A ^ A ; 1 : 1 where A is the magnitude of vector A and so it has unit and dimension, and ^ A is a dimensionless unit vector with a unity magnitude having the direction of A . Thus ^ A A = A . 1 A vector quantity may be represented graphically by an arrow-tipped line seg-ment. The length of the arrow represents the magnitude of the vector, and the direction of the arrow is that of the vector, as shown in Fig.
eBook - ePub- Tai-Ran Hsu(Author)
- 2018(Publication Date)
- Wiley(Publisher)
Chapter 3 Vectors and Vector CalculusChapter Learning Objectives
- Recap the distinction between scalar and vector quantities in engineering analysis.
- Learn vector calculus and its applications in engineering analysis.
- Learn to manipulate expressions of vectors and vector functions.
- Refresh vector algebra.
- Learn the dot and cross products of vectors and their physical meanings.
- Learn about derivatives, gradient, divergence, and curl in vector calculus.
- Learn to apply vector calculus in engineering analysis.
- Learn to apply vector calculus in rigid body dynamics in rectilinear and plane curvilinear motion along paths and in both rectangular and cylindrical polar coordinate systems.
3.1 Vector and Scalar Quantities
In Section 2.2.3 , we introduced functions that represent physical quantities in engineering analyses, and whose values vary with the values of the associated independent variables in space (x, y, z) in a rectangular coordinate system) and time (t). These quantities are called as scalar quantities.There is another group of physical quantities for which not only the magnitude but also the position and the direction are significant and must be represented.. These are called vector quantities. Thus, a speed of 80 km/h of a moving car is a scalar quantity, but a velocity of 80 km/h implies that the car is traveling in specific direction on the road at this speed, so that it is a vector quantity. Engineering analyses involving vectorial quantities will require the use of vector calculus, which will be described later in Section 3.5 .A vector, as stated, is characterized by both its magnitude and its direction
eBook - PDF- Bhag Singh Guru, Hüseyin R. Hiziroglu(Authors)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
2 Vector analysis 2.1 Introduction ................................. Knowledge of vector algebra and vector calculus is essential in de-veloping the concepts of electromagnetic field theory. The widespread acceptance of vectors in electromagnetic field theory is due in part to the fact that they provide compact mathematical representations of complicated phenomena and allow for easy visualization and manipula-tion. The ever-increasing number of textbooks on the subject are further evidence of the popularity of vectors. As you will see in subsequent chapters, a single equation in vector form is sufficient to represent up to three scalar equations. Although a complete discussion of vectors is not within the scope of this text, some of the vector operations that will play a prominent role in our discussion of electromagnetic field theory are introduced in this chapter. We begin our discussion by defining scalar and vector quantities. 2.2 Scalar and vector quantities ................................. Most of the quantities encountered in electromagnetic fields can easily be divided into two classes, scalars and vectors. 2.2.1 Scalar A physical quantity that can be completely described by its magnitude is called a scalar . Some examples of scalar quantities are mass, time, temperature, work, and electric charge. Each of these quantities is com-pletely describable by a single number. A temperature of 20 ◦ C, a mass of 100 grams, and a charge of 0.5 coulomb are examples of scalars. In fact, all real numbers are scalars. 2.2.2 Vector A physical quantity having a magnitude as well as a direction is called a vector . Force, velocity, torque, electric field, and acceleration are vector quantities. 14 15 2.3 Vector operations A vector quantity is graphically depicted by a line segment equal to its magnitude according to a convenient scale, and the direction is indicated by means of an arrow, as shown in Figure 2.1a.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Vector operations also have numerous generalizations in other branches of physics. Chapter 2 | Vectors 43 2.1 | Scalars and Vectors Learning Objectives By the end of this section, you will be able to: • Describe the difference between vector and scalar quantities. • Identify the magnitude and direction of a vector. • Explain the effect of multiplying a vector quantity by a scalar. • Describe how one-dimensional vector quantities are added or subtracted. • Explain the geometric construction for the addition or subtraction of vectors in a plane. • Distinguish between a vector equation and a scalar equation. Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, “a class period lasts 50 min” or “the gas tank in my car holds 65 L” or “the distance between two posts is 100 m.” A physical quantity that can be specified completely in this manner is called a scalar quantity. Scalar is a synonym of “number.” Time, mass, distance, length, volume, temperature, and energy are examples of scalar quantities. Scalar quantities that have the same physical units can be added or subtracted according to the usual rules of algebra for numbers. For example, a class ending 10 min earlier than 50 min lasts 50 min − 10 min = 40 min . Similarly, a 60-cal serving of corn followed by a 200-cal serving of donuts gives 60 cal + 200 cal = 260 cal of energy. When we multiply a scalar quantity by a number, we obtain the same scalar quantity but with a larger (or smaller) value. For example, if yesterday’s breakfast had 200 cal of energy and today’s breakfast has four times as much energy as it had yesterday, then today’s breakfast has 4(200 cal) = 800 cal of energy. Two scalar quantities can also be multiplied or divided by each other to form a derived scalar quantity.
eBook - PDFUnderstanding Geometric Algebra
Hamilton, Grassmann, and Clifford for Computer Vision and Graphics
- Kenichi Kanatani(Author)
- 2015(Publication Date)
- A K Peters/CRC Press(Publisher)
C H A P T E R 2 3D Euclidean Geometry 3D Euclidean geometry is usually described in terms of numerical vectors, i.e., column and row vectors. In this chapter, a vector is a geometric object equipped with direction and magnitude, not an array of numbers. Here, we present an “algebraic” description of 3D Euclidean geometry, representing objects by “symbols” and defining “operations” on them. From this viewpoint, we introduce the inner, the outer, and the scalar triple products of vectors. Then, we list expressions and relationships involving rotations, projections, lines, and planes. This chapter provides a basis of all the subsequent chapters. 2.1 VECTORS A vector is a geometric object that has “direction” and “magnitude”; one can imagine it as an “arrow” in space. Vectors represent the following quantities and properties: Displacements, velocities, and force Vectors specify in which direction, over what distance, and at what velocity things move or what force is acting. We are interested only in their directions and magnitudes; we are not concerned with the location of the starting point. Such vectors are called free vectors . Directions in space Vectors indicate orientations of lines and surface normals to planes. Only directions are important; magnitudes are ignored. Such vectors are called direction vectors . Since their magnitudes are irrelevant, they are usually multiplied by appropriate numbers to unit vectors with unit magnitude. Positions in space We fix a special point O , called the origin , and define the positions of points by their displacements from the origin O . Such vectors are called position vectors . Vectors whose starting points are specified are said to be bound . Positions vectors are bound to the origin O . The study of using and manipulating “vectors” for such different quantities and rela-tionships is called vector calculus , providing the basis of geometry, classical mechanics, and electrodynamics.
eBook - PDFPrinciples of Continuum Mechanics
Conservation and Balance Laws with Applications
- J. N. Reddy(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
A study of physical phenomena by means of vectors and tensors may lead to a deeper understanding of the problem in addition to bringing simplicity and versatility into the analysis. This chapter is dedicated to the study of algebra and calculus of physical vectors and tensors, as needed in the subsequent study. 2.2 Definition of a Vector The quantities encountered in analytical description of physical phenomena may be classified into the following two groups according to the information needed to specify them completely: scalars and nonscalars (see Table 2.1). The scalars are given by a single number. Nonscalars have not only a magnitude specified, but also additional information, such as direction. Nonscalars that obey certain rules 14 VECTORS AND TENSORS Table 2.1 Classification of mathematical quantities Scalars Nonscalars Mass Force Temperature Moment Time Stress Volume Acceleration Length Displacement (such as the parallelogram law of addition) are called vectors . Not all nonscalar quantities are vectors, unless they obey certain rules as discussed next. A physical vector is often shown as a directed line segment with an arrowhead at the end of the line, as shown in Fig. 2.1. The length of the line represents the magnitude of the vector and the arrow indicates the direction. In written or typed material, it is customary to place an arrow over the letter denoting the vector, such as ~ A . In printed material, the letter used for the vector is commonly denoted by a boldface letter, A , such as used in this study. The magnitude of the vector A is denoted by | A | , k A k , or A . Magnitude of a vector is a scalar. Actual computation of the magnitude of a general vector requires the notion of a “norm” of a vector. The “dot product” of a physical vector with itself gives the square of the length, as will be discussed in Section 2.3.5. Thus, mathematically we can only find the square of the magnitude of a vector.
eBook - PDFMathematics for Engineering, Technology and Computing Science
The Commonwealth and International Library: Electrical Engineering Division
- Hedley G. Martin, N. Hiller(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
The rest of the 294 VECTOR ANALYSIS 295 chapter deals with this aspect, the true vector analysis, but the treatment is concerned with a proper understanding of funda-mental ideas rather than with applications. 8.2. DEFINITIONS Quantities that possess magnitude only are completely de-fined by a real number and the unit to which the number refers. Such quantities are called scalars and familiar examples are volume, mass, temperature, charge and electric potential, as well as numbers themselves which are not necessarily related to physical concepts. Scalars are combined in various ways by the familiar operations of elementary algebra. Many quantities require for their definition a real number and a direction. Provided they satisfy certain conditions such quantities are called vectors and examples are velocity, acceler-ation, electric and magnetic intensity. Vectors are combined according to the laws of vector algebra, which take account of the directional property in ways which are consistent with how directed quantities combine in practice. Although vectors are said to be added and multiplied the processes are not simply those of scalar algebra, from which the terminology and much of the notation comes, and care should be taken not to identify familiar ideas with the new ones of vector algebra. A vector may be represented by or considered as a directed line segment AB, shown in Fig. 8.1. The vector which the seg- ment represents may be denoted by or by a single letter in boldface type, such as v; in manuscript a vector quantity is indicated by underlining a letter, as in v. The end points A and B of the segment shown are called the initial and terminal points respectively. The magnitude of v, always positive, is indicated by the length of the segment and may be denoted in various ways, such as by ÁB , I v or v. The direction of v is defined in relation to a frame of reference, which is taken here H GM : - METCS 20
- Maxum, Bernard(Authors)
- 2004(Publication Date)
2-1 Chapter 2 Vector Algebra Review The purpose of this chapter is to review some of the salient operations involving scalar and vector fields and to broaden these concepts to dyadics and tensors in general. Here we briefly discuss variant and invariant scalars, the concept of scalar and vector fields, and the utility of phasor forms of these quantities. Classical arithmetic vector operations of addition, subtraction, and dot and cross products are discussed along with physical applications of these. The direct vector-vector product is mentioned in Section 2.4.3 as having a dyadic resultant; however, the details of this process are left to later chapters. The basic building blocks of open and closed line and surface integrals of vector fields are discussed. These are essential for both the definitions of vector differential operators, covered in Chapter 4, and the integral forms that shape the basis of divergence, Stokes’, and Green’s theorems covered in Chapter 5. Other highly useful applications of dot- and cross-product operations conclude the sections of this chapter. These are vector field direction lines and equivalue surfaces of scalar fields. 2.1 Variant and Invariant Scalars A quantity is said to be a scalar if it has only magnitude, that is, no inherent direction. Quantities such as time, mass, distance, temperature, entropy, energy, electric potential, and pressure have a value at every position in space but lack directionality. These are scalars. Because such quantities are independent of the orientation of a coordinate system, they are called invariant scalars . Coordinates of a point and components of a vector are also scalars; however, these quantities change with coordinate displacements and rotations and therefore are variant scalars. 2.2 Scalar Fields In general, scalar fields are quantities that can be represented by functions of space and time.
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Summary › Definitions Scalar quantities are those that have only a numerical value and no associated direction. Vector quantities have both magnitude and direction and obey the laws of vector addition. The magnitude of a vector is always a positive number. › Concepts and Principles When two or more vectors are added together, they must all have the same units and they all must be the same type of quantity. We can add two vectors A S and B S graphically. In this method (Fig. 3.6), the resultant vector R S 5 A S 1 B S runs from the tail of A S to the tip of B S . A second method of adding vectors involves components of the vec- tors. The x component A x of the vector A S is equal to the projection of A S along the x axis of a coordinate system, where A x 5 A cos u. The y component A y of A S is the projection of A S along the y axis, where A y 5 A sin u. If a vector A S has an x component A x and a y compo- nent A y , the vector can be expressed in unit-vector form as A S 5 A x i ⁄ 1 A y j ⁄ . In this notation, i ⁄ is a unit vector point- ing in the positive x direction and j ⁄ is a unit vector point- ing in the positive y direction. Because i ⁄ and j ⁄ are unit vectors, u i ⁄ u 5 u j ⁄ u 5 1. We can find the resultant of two or more vectors by resolving all vectors into their x and y components, adding their resultant x and y components, and then using the Pythagorean theorem to find the magnitude of the resultant vector.
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