Physics
Scalar and Vector
Scalars are quantities that are fully described by their magnitude, such as mass or temperature. Vectors, on the other hand, have both magnitude and direction, like velocity or force. In physics, understanding the distinction between scalars and vectors is crucial for accurately describing and analyzing physical phenomena.
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11 Key excerpts on "Scalar and Vector"
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 - 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.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Scalars, such as temperature, have magnitude only. They are specified by a num- ber with a unit (10°C) and obey the rules of arithmetic and ordinary algebra. Vectors, such as displacement, have both magnitude and direction (5 m, north) and obey the rules of vector algebra. 2. Two vectors a → and b → may be added geometrically by drawing them to a com- mon scale and placing them head to tail. The vector connecting the tail of the first to the head of the second is the vector sum s → . To subtract b → from a → , reverse the direction of b → to get ‒ b → ; then add ‒ b → to a → . Vector addition is com- mutative and obeys the associative law. 3. The (scalar) components a x and a y of any two-dimensional vector a → along the coordinate axes are found by dropping perpendicular lines from the ends of a → onto the coordinate axes. The components are given by a x = a cos θ and a y = a sin θ, where θ is the angle between the positive direction of the x axis and the direc- tion of a → . The algebraic sign of a component indicates its direction along the associated axis. Given its components, we can find the magnitude and orien- tation of the vector a → with a = √ _______ a x 2 + a y 2 and tan θ = a y __ a x . 41 42 CHAPTER 3 Vectors A vector has magnitude as well as direction, and vectors follow certain (vector) rules of combination, which we examine in this chapter. A vector quantity is a quantity that has both a magnitude and a direction and thus can be represented with a vector. Some physical quantities that are vector quantities are displacement, velocity, and acceleration. You will see many more throughout this book, so learn- ing the rules of vector combination now will help you greatly in later chapters. Not all physical quantities involve a direction. Temperature, pressure, energy, mass, and time, for example, do not “point” in the spatial sense. We call such quantities scalars, and we deal with them by the rules of ordinary algebra.
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- Daniel A. Fleisch(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
To understand how vectors are different from other entities, it may help to consider the nature of some things that are clearly not vectors. Think about the temperature in the room in which you’re sitting – at each point in the room, the temperature has a value, which you can represent by a single number. That value may well be different from the value at other locations, but at any given point the temperature can be represented by a single number, the magnitude. Such magnitude-only quantities have been called “scalars” ever since W.R. Hamilton referred to them as “all values contained on the one scale of progres-sion of numbers from negative to positive infinity.” 4 Thus A scalar is the mathematical representation of a physical entity that may be characterized by magnitude only. Other examples of scalar quantities include mass, charge, energy, and speed (defined as the magnitude of the velocity vector). It is worth noting that the change in temperature over a region of space does have both magnitude and direction and may therefore be represented by a vector, so it’s possible to pro-duce vectors from groups of scalars. You can read about just such a vector (called the “gradient” of a scalar field) in Chapter 2 . Since scalars can be represented by magnitude only (single numbers) and vectors by magnitude and direction (three numbers in three-dimensional space), you might suspect that there are other entities involving magnitude and directions that are more complex than vectors (that is, requiring more numbers than the number of spatial dimensions). Indeed there are, and such entities are called “tensors.” 5 You can read about tensors in the last three chapters of this book, but for now this simple definition will suffice: 4 W.R.
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.
- 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
eBook - PDF- Nelson Bolívar(Author)
- 2020(Publication Date)
- Arcler Press(Publisher)
The small curve segment can be approximated around the point P with an arc of the osculating circle and consider the point as moving on arc with the angular velocity ω = υ / R . In conclusion, two components of acceleration are as: 2 t n dv v a ,a . dt R = = (52) 3.13. VECTORS, PSEUDOVECTORS, SCALARS, AND PSEUDOSCALARS Previously, the vector is defined as a well-ordered triple of the real numbers that under the rotations of reference frame changes in same way as the triplet characterizing the position vector (Kamiński et al., 1997; Maris and Roberts, 1998). Previously, a scalar quantity is introduced, dot product of the two vectors. It is seen that scalar quantity is same in the two reference frames differing for the rotation of axes. Indeed, generally, a quantity, by definition, is scalar if the quantity is invariant under the change of reference frame (Hsiao et al., 2000; Chou, 2009). Hence, both Scalar and Vector properties are articulated in terms of changes between the reference frames. Let’s now contemplate the behaviors of both these quantities under inversion of the axes. It is known as a parity operation. It leads from the left-handed frame to right-handed one (McConnell, 1958; Bennhold and Wright, 1987). Consider the variation properties of the physical quantities. The quantity can be scalar or can be pseudoscalar. Both are invariant under the rotations General Physics 80 but the earlier is invariant under the parity operation, the latter varies sign, whereas keeping the absolute value (Deo and Bisoi, 1974; Chiu et al., 2006). The dot product of the two vectors is scalar; the scalar triple product is pseudoscalar. This is immediately evident seeing that under the inversion of axes all the 3 vector factors transform sign. The quantity can be vector or pseudovector (Norton and Watson, 1958; Sugawara and Okubo, 1960).
eBook - PDF- Harold Jeffreys, Bertha Jeffreys(Authors)
- 1999(Publication Date)
- Cambridge University Press(Publisher)
What we shall do in the present chapter is to show the two methods, as far as possible, side be- side. Ability to translate from either language to the other, or to the expanded form, is an absolute necessity in understanding modern physical literature. It is often useful to visualize a vector as a displacement vector, and while as a matter of definition we make a clear distinction between a general vector and a displacement vector, we shall frequently speak of a general vector in geometrical terms: e.g. the angle between two vectors A and B means strictly 'the angle between the displacement vectors representing A and B according to some specified scale'; * two perpendicular vectors A and B* means 'two vectors A and B such that the displacements representing them are perpendicular'. The use of this analogy is unnecessary in suffix notation, analytical definitions being provided. 2*033. Null vector. A null or zero vector is one whose modulus is zero. 2*034. Direction vectors. A vector of modulus 1 (a number) in the direction of a vector A is called a unit or direction vector in that direction. Its components are evidently l i9 the direction cosines of the direction of A with regard to the coordinate axes. In particular we shall denote direction vectors in the directions of the axes by e (1) , e (2 ), e^ respectively; that is, e (1) = (1,0,0), e (2) = (0,1,0), e (3) = (0,0,1). The use of the brackets round the suffix is to emphasize that it does not denote a com- ponent, but a particular vector. Any vector A may be written as ^ I e ( l ) + ^2 e (2) + ^3*3(3)- Some books denote direction vectors parallel to the axes by i f j, k and write A = A x i + A y j + A 8 k. 2-04. Linearly dependent or coplanar vectors. If there is a relation aA + fiB + yC = 0 (1) between three vectors, where a.fi^y are real numbers (not all zero), then A, B, C are said to be linearly dependent.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
One second is the time for a certain type of electromagnetic wave emitted by cesium-133 atoms to undergo 9 192 631 770 wave cycles. 1.3 The Role of Units in Problem Solving To convert a number from one unit to another, multiply the number by the ratio of the two units. For instance, to convert 979 meters to feet, multiply 979 meters by the factor (3.281 foot/1 meter). The dimension of a quantity represents its physical nature and the type of unit used to specify it. Three such dimensions are length [L], mass [M], time [T]. Dimensional analysis is a method for checking mathematical relations for the consistency of their dimensions. 1.4 Trigonometry The sine, cosine, and tangent functions of an angle u are defined in terms of a right triangle that contains u, as in Equations 1.1–1.3, where h o and h a are, respectively, the lengths of the sides opposite and adjacent to the angle u, and h is the length of the hypotenuse. The inverse sine, inverse cosine, and inverse tangent functions are given in Equations 1.4–1.6. The Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides, according to Equation 1.7. 1.5 Scalars and Vectors A scalar quantity is described by its size, which is also called its magnitude. A vector quantity has both a magnitude and a direction. Vectors are often represented by arrows, the length of the arrow being proportional to the magnitude of the vector and the direction of the arrow indicating the direction of the vector. 1.6 Vector Addition and Subtraction One procedure for adding vectors utilizes a graphical technique, in which the vectors to be added are arranged in a tail-to-head fashion. The resultant vector is drawn from the tail of the first vector to the head of the last vector. The subtraction of a vector is treated as the addition of a vector that has been multiplied by a scalar factor of 21.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The scalar component A y is defined in a similar manner. The following table shows an example of vector and scalar components: In this text, when we use the term “component,” we will be referring to a scalar component, unless otherwise indicated. Another method of expressing vector components is to use unit vectors. A unit vector is a vector that has a magnitude of 1, but no dimensions. We will use a caret (^) to distinguish it from other vectors. Thus, x ˆ is a dimensionless unit vector of length l that points in the positive x direction, and y ˆ is a dimensionless unit vector of length l that points in the positive y direction. These unit vectors are illustrated in Figure 1.19. With the aid of unit vectors, the vector com- ponents of an arbitrary vector A → can be written as A → x = A x x ˆ and A → y = A y y ˆ, where A x and A y are its scalar components (see the drawing and the third column of the table above). The vector A → is then written as A → = A x x ˆ + A y y ˆ. Resolving a Vector into Its Components If the magnitude and direction of a vector are known, it is possible to find the components of the vector. The process of finding the components is called “resolving the vector into its components.” As Example 8 illustrates, this process can be carried out with the aid of trigo- nometry, because the two perpendicular vector components and the original vector form a right triangle. A y +y +x A x A θ FIGURE 1.17 This alternative way of drawing the vector A → and its vector components is completely equivalent to that shown in Interactive Figure 1.16. +y +x +y´ +x´ A x A´ y A´ x A A y FIGURE 1.18 The vector components of the vector depend on the orientation of the axes used as a reference. x y A x x A y y +x +y FIGURE 1.19 The dimensionless unit vectors x ˆ and y ˆ have magnitudes equal to 1, and they point in the +x and +y directions, respectively. Expressed in terms of unit vectors, the vector components of the vector A → are A x x ˆ and A y y ˆ.
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