Physics
Vector Algebra
Vector algebra is a branch of mathematics that deals with operations and manipulations involving vectors, which are quantities that have both magnitude and direction. In physics, vector algebra is used to represent physical quantities such as force, velocity, and acceleration. It involves addition, subtraction, and multiplication of vectors, as well as concepts like dot and cross products.
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12 Key excerpts on "Vector Algebra"
- Gregory J. Gbur(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
1 Vector Algebra 1.1 Preliminaries In introductory physics, we often deal with physical quantities that can be described by a single number. The temperature of a heated body, the mass of an object, and the electric potential of an insulated metal sphere are all examples of such scalar quantities. Descriptions of physical phenomena are not always (indeed, rarely) that simple, however, and often we must use multiple, but related, numbers to offer a complete description of an effect. The next level of complexity is the introduction of vector quantities. A vector may be described as a conceptual object having both magnitude and direction. Graphically, vectors can be represented by an arrow: The length of the arrow is the magnitude of the vector, and the direction of the arrow indicates the direction of the vector. Examples of vectors in elementary physics include displacement, velocity, force, momen- tum, and angular momentum, though the concept can be extended to more complicated and abstract systems. Algebraically, we will usually represent vectors by boldface characters, i.e. F for force, v for velocity, and so on. It is worth noting at this point that the word “vector” is used in mathematics with some- what broader meaning. In mathematics, a vector space is defined quite generally as a set of elements (called vectors) together with rules relating to their addition and scalar mul- tiplication of vectors. In this sense, the set of real numbers form a vector space, as does any ordered set of numbers, including matrices, to be discussed in Chapter 4, and complex numbers, to be discussed in Chapter 9. For most of this chapter we reserve the term “vector” for quantities which possess magnitude and direction in three-dimensional space, and are independent of the specific choice of coordinate system in a manner to be discussed in 1 2 Vector Algebra A A B B C Figure 1.1 The parallelogram law of vector addition.
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
9 Vector Algebra This chapter introduces space vectors and their manipulation. Firstly we deal with the description and algebra of vectors, then we consider how vectors may be used to describe lines, planes and spheres, and finally we look at the practical use of vectors in finding distances. The calculus of vectors will be developed in a later chapter; this chapter gives only some basic rules. 9.1 Scalars and vectors • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • The simplest kind of physical quantity is one that can be completely specified by its magnitude, a single number, together with the units in which it is measured. Such a quantity is called a scalar , and examples include temperature, time and density. A vector is a quantity that requires both a magnitude ( ≥ 0) and a direction in space to specify it completely; we may think of it as an arrow in space. A familiar example is force, which has a magnitude (strength) measured in newtons and a direction of application. The large number of vectors that are used to describe the physical world include velocity, displacement, momentum and electric field. Vectors can also be used to describe quantities such as angular momentum and surface elements (a surface element has a magnitude, defined by its area, and a direction defined by the normal to its tangent plane); in such cases their definitions may seem somewhat arbitrary (though in fact they are standard) and not as physically intuitive as for vectors such as force. A vector is denoted by bold type, the convention of this book, or by underlining, the latter being much used in handwritten work. This chapter considers basic Vector Algebra and illustrates just how powerful vector analysis can be. All the techniques are presented for three-dimensional space but most can be readily extended to more dimensions.
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 - PDF- Tom M. Apostol(Author)
- 2019(Publication Date)
- Wiley(Publisher)
W. Gibbs (1839–1903) and O. Heaviside (1850–1925), and a new subject called Vector Algebra sprang into being. It was soon realized that vectors are the ideal tools for the exposition and simplification of many important ideas in geometry and physics. In this chapter we propose to discuss the elements of Vector Algebra. Applications to analytic geometry are given in Chapter 13. In Chapter 14 Vector Algebra is combined with the methods of calculus, and applications are given to both geometry and mechanics. There are essentially three different ways to introduce Vector Algebra: geometrically, analyt- ically, and axiomatically. In the geometric approach, vectors are represented by directed line segments, or arrows. Algebraic operations on vectors, such as addition, subtraction, and multipli- cation by real numbers, are defined and studied by geometric methods. In the analytic approach, vectors and vector operations are described entirely in terms of num- bers, called components. Properties of the vector operations are then deduced from corresponding properties of numbers. The analytic description of vectors arises naturally from the geometric description as soon as a coordinate system is introduced. 445 446 Vector Algebra In the axiomatic approach, no attempt is made to describe the nature of a vector or of the algebraic operations on vectors. Instead, vectors and vector operations are thought of as unde- fined concepts of which we know nothing except that they satisfy a certain set of axioms. Such an algebraic system, with appropriate axioms, is called a linear space or a linear vector space. Examples of linear spaces occur in all branches of mathematics, and we will study many of them in Chapter 15. The algebra of directed line segments and the algebra of vectors described by components are merely two examples of linear spaces.
- Jeremy Dunning-Davies(Author)
- 2003(Publication Date)
- Woodhead Publishing(Publisher)
Chapter 6 Vector Algebra 6.1 INTRODUCTION For any given physical system, many of its physical properties may be classi-fied as either scalars or vectors. The difference between these two classes is intuitive but, roughly speaking, a scalar is a quantity possessing magnitude, whereas a vector is a quantity possessing both magnitude and direction. Any scalar is specified completely by a real number, and the laws of addition and multiplication obeyed by scalars are the same as those for real numbers. For real numbers a, b, c . .., these laws are: Commutative laws. a + b = b + a ; ab = ba . Associative laws. a + (6 + c) = (a + b) + c; a(bc) = (o6)c Distributive law. a(b + c) = ab + ac. Also, subtraction and division may be defined in terms of addition and multi-plication as follows: c - a = c + (-a) where (-a) is defined by 6 + α = 0 φ 6 = (-a) c/a = c a' 1 where a -1 is defined by ba= 1 Φ 6 = ο ' . Examples of scalars are length, speed, time, and energy, while displace-ment, velocity, acceleration, force, and momentum are all examples of vectors. Sec. 6.2] Representation of a Vector 155 Consider displacement. If a particle is displaced from A to Β (see Fig. 6.1), then the displacement is given not only by the length AB but also its direction. Β A - ^ Fig. 6.1 Length is actually the magnitude of displacement. 6.2 REPRESENTATION OF A VECTOR A vector is represented by a line of the required magnitude pointing in the required direction. Vectors are denoted by boldface type, a, b, etc. usually. Also, the vector represented by the displacement from Ο to A may be denoted by OA. Hence, if a is represented by the displacement from Ο to A a = Ok*. Then, if a = OA, the vector -a may be defined by -a = AO* that is, a and -a have equal magnitude but opposite direction. There are two general categories of vectors: fixed vectors and free vectors.
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 - PDFDynamics of the Atmosphere
A Course in Theoretical Meteorology
- Wilford Zdunkowski, Andreas Bott(Authors)
- 2003(Publication Date)
- Cambridge University Press(Publisher)
Part 1 Mathematical tools M1 Algebra of vectors M1.1 Basic concepts and definitions A scalar is a quantity that is specified by its sign and by its magnitude. Examples are temperature, the specific volume, and the humidity of the air. Scalars will be written using Latin or Greek letters such as a, b, . . . , A, B, . . . , α, β, . . . . A vector requires for its complete characterization the specification of magnitude and direction. Examples are the velocity vector and the force vector. A vector will be represented by a boldfaced letter such as a , b , . . . , A , B , . . . . A unit vector is a vector of prescribed direction and of magnitude 1. Employing the unit vector e A , the arbitrary vector A can be written as A = | A | e A = A e A =⇒ e A = A | A | (M1 . 1) Two vectors A and B are equal if they have the same magnitude and direction regardless of the position of their initial points, that is | A | = | B | and e A = e B . Two vectors are collinear if they are parallel or antiparallel. Three vectors that lie in the same plane are called coplanar . Two vectors always lie in the same plane since they define the plane. The following rules are valid: the commutative law : A ± B = B ± A , A α = α A the associative law : A + ( B + C ) = ( A + B ) + C , α ( β A ) = ( αβ ) A the distributive law : ( α + β ) A = α A + β A (M1 . 2) The concept of linear dependence of a set of vectors a 1 , a 2 , . . . , a N is closely connected with the dimensionality of space. The following definition applies: A set of N vectors a 1 , a 2 , . . . , a N of the same dimension is linearly dependent if there exists a set of numbers α 1 , α 2 , . . . , α N , not all of which are zero, such that α 1 a 1 + α 2 a 2 + · · · + α N a N = 0 (M1 . 3) 3 4 Algebra of vectors Fig. M1.1 Linear vector spaces: (a) one-dimensional, (b) two-dimensional, and (c) three-dimensional. If no such numbers exist, the vectors a 1 , a 2 , . .
eBook - PDF- Harold Josephs, Ronald Huston(Authors)
- 2002(Publication Date)
- CRC Press(Publisher)
15 2 Review of Vector Algebra 2.1 Introduction In Chapter 1, we reviewed the basic concepts of vectors. We considered vectors, scalars, and the multiplication of vectors and scalars. We also examined zero vectors and unit vectors. In this chapter, we will build upon these ideas as we develop a review of Vector Algebra. Specifically, we will review the concepts of vector equality, vector addition, and vector multiplication. We will also review the concepts of reference frames and unit vector sets. Finally, we will review the elementary procedures of matrix algebra. 2.2 Equality of Vectors, Fixed and Free Vectors Recall that the characteristics of a vector are its magnitude , its orientation , and its sense . Indeed, we could say that a vector is defined by its characteristics. The concept of vector equality follows from this definition: Specifically, two vectors are equal if (and only if) they have the same characteristics. Vector equality is fundamental to the development of Vector Algebra. For example, if vectors are equal, they may be interchanged in vector equations, which enables us to simplify expressions. It should be noted, however, that vector equality does not necessarily denote physical equality, particularly when the vec-tors model physical quantities. This occurs, for example, with forces. We will explore this concept later. Two fundamental ideas useful in relating mathematical and physical quantities are the concepts of fixed and free vectors. A fixed vector has its location restricted to a line fixed in space. To illustrate this, consider the fixed line L as shown in Figure 2.2.1. Let v be a vector whose location is restricted to L , and let the location of v along L be arbitrary. Then v is a fixed vector. Because the location of v along L is arbitrary, v might even be called a sliding vector . Alternatively, a free vector is a vector that may be placed anywhere in space if its characteristics are maintained.
eBook - PDF- J. N. Reddy(Author)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
Like physical vectors, tensors are more general objects that are endowed with a magni- tude and multiple direction(s) and satisfy rules of tensor addition and scalar mul- tiplication. In fact, physical vectors are often termed the first-order tensors. As will be shown shortly, the specification of a stress component (i.e., force per unit area) requires a magnitude and two directions – one normal to the plane on which the stress component is measured and the other is its direction – to specify it uniquely. 8 2.2 Vector Algebra 9 This chapter is dedicated to a review of algebra and calculus of physical vectors and tensors. Those who are familiar with the material covered in any of the sections may skip them and go to the next section or Chapter 3. 2.2 Vector Algebra In this section, we present a review of the formal definition of a geometric (or phys- ical) vector, discuss various products of vectors and physically interpret them, in- troduce index notation to simplify representations of vectors in terms of their com- ponents as well as vector operations, and develop transformation equations among the components of a vector expressed in two different coordinate systems. Many of these concepts, with the exception of the index notation, may be familiar to most students of engineering, physics, and mathematics and may be skipped. 2.2.1 Definition of a Vector The quantities encountered in analytical description of physical phenomena may be classified into two groups according to the information needed to specify them completely: scalars and nonscalars. The scalars are given by a single number. Non- scalars have not only a magnitude specified but also additional information, such as direction. Nonscalars that obey certain rules (such as the parallelogram law of addition) are called vectors. Not all nonscalar quantities are vectors (e.g., a finite rotation is not a vector).
eBook - PDFEngineering Electromagnetics
Pergamon Unified Engineering Series
- David T. Thomas, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
1 Vector Analysis and Coordinates INTRODUCTION This chapter on vector analysis and coordinates is included because of; (1) the tremendous importance of vectors in electromagnetics, and (2) the absence of a course on vector analysis elsewhere in most college curricula. Even if a separate vector analysis course exists in your university, a brief review should be included in any course in electromagnetics to make certain that the necessary material is understood. Vector analysis is a separate chapter because of its reference nature. Vector properties must be distinguishable from electromagnetic properties, and must be accessible to the student. DEFINITIONS AND ALGEBRA A scalar field or function assigns a value for each point in space, or at least a region of space. Typical examples of scalar fields are temperature, J; pressure, p charge, Q time, V, and mass, m. Notice in each example a single quantity, independent of direction, is assigned for each space point of interest. Each of you should be familiar with scalars, as the functions of calculus are scalars of a one dimensional variable. Here, we must consider scalars of three space variables which complicates matters somewhat. A vector field or function on the other hand assigns both a value or magnitude and a direction for each point in the space of interest. Examples include velocity, v; force, F; displacement, s; and current density, J. In writing vectors the notation normally used to indicate vectors is to draw a line above the symbol thusly,yi. A caution is extended regarding the common practice of drawing vectors in space with their magnitude indicated by the length. This does not imply the vector extends in space, but merely is a device used to show magnitude! Invariant Single Valued Functions All physical scalars and vectors are single valued and invariant. Both wifl be very important in proving other properties and solving problems. To say a funç-1
eBook - ePubVector Analysis and Cartesian Tensors
Third Edition
- Donald Edward Bourne(Author)
- 2018(Publication Date)
- Chapman and Hall/CRC(Publisher)
Scalar and Vector Algebra 2 2.1 SCALARSAny mathematical entity or any property of a physical system which can be represented by a single real number is called a scalar. In the case of a physical property, the single real number will be a measure of the property in some chosen units (e.g. kilogrammes, metres, seconds).Particular examples of scalars are: (i) the modulus of a complex number; (ii) mass; (iii) volume; (iv) temperature. Note that real numbers are themselves scalars.Single letters printed in italics (such as a, b, c, etc.) will be used to denote real numbers representing scalars. For convenience statements like ‘let a be a real number representing a scalar’ will be abbreviated to ‘let a be a scalar’.Equality of scalars Two scalars (measured in the same units if they are physical properties) are said to be equal if the real numbers representing them are equal.It will be assumed throughout this book that in the case of physical entities the same units are used on both sides of any equality sign.Scalar addition, subtraction, multiplication and divisionThe sum of two scalars is defined as the sum of the real numbers representing them. Similarly, scalar subtraction, multiplication and division are defined by the corresponding operations on the representative numbers. In the case of physical scalars, the operations of addition and subtraction are physically meaningful only for similar scalars such as two lengths or two masses.Some care is necessary in the matter of units. For example, if a, b are two physical scalars it is meaningful to say their sum is a + b only if the units of measurement are the same.Again, consider the equationT =giving the kinetic energy T of a particle of mass m travelling with speed v. If T has the value 30 kg m2 s−2 and v has the value 0.1 km s−1 , then to calculate m = 2T/v2 consistent units for length and time must first be introduced. Thus, converting the given speed to m s−1 we find v has the value 100 m s−1 . Hence the value of m1 2mv 2
- P C Kendall, D.E. Bourne(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
2 Scalar and Vector Algebra2.1 Scalars
Any mathematical entity or any property of a physical system which can be represented by a single real number is called a scalar . In the case of a physical property, the single real number will be a measure of the property in some chosen units (e.g. kilogrammes, metres, seconds).Particular examples of scalars are: (i) the modulus of a complex number; (ii) mass; (iii) volume; (iv) temperature. Note that real numbers are themselves scalars.Single letters printed in italics (such as a, b, c , etc.) will be used to denote real numbers representing scalars. For convenience statements like ‘let a be a real number representing a scalar’ will be abbreviated to ‘let a be a scalar’.Equality of scalars Two scalars (measured in the same units if they are physical properties) are said to be equal if the real numbers representing them are equal.It will be assumed throughout this book that in the case of physical entities the same units are used on both sides of any equality sign .Scalar addition, subtraction, multiplication and divisionThe sum of two scalars is defined as the sum of the real numbers representing them. Similarly, scalar subtraction, multiplication and division are defined by the corresponding operations on the representative numbers. In the case of physical scalars, the operations of addition and subtraction are physically meaningful only for similar scalars such as two lengths or two masses.Some care is necessary in the matter of units. For example, if a, b are two physical scalars it is meaningful to say their sum is a + b only if the units of measurement are the same.Again, consider the equationT =giving the kinetic energy T of a particle of mass m travelling with speed v . If T has the value 30 kg m2 s-2 and v has the value 0.1 km s-1 , then to calculate m -2T /v 2 consistent units for length and time must first be introduced. Thus, converting the given speed to m s-1 we find v has the value 100 m s-l . Hence the value of m1 2mv 2
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