Mathematics
Vectors
Vectors are mathematical objects that have both magnitude and direction. They are often represented as arrows in space, with the length of the arrow indicating the magnitude and the direction indicating the direction. Vectors are used in various mathematical fields, including geometry, physics, and engineering, to represent quantities such as force, velocity, and displacement.
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12 Key excerpts on "Vectors"
eBook - PDF- Daniel A. Fleisch(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
1 Vectors 1.1 Definitions (basic) There are many ways to define a vector. For starters, here’s the most basic: A vector is the mathematical representation of a physical entity that may be characterized by size (or “magnitude”) and direction. In keeping with this definition, speed (how fast an object is going) is not rep-resented by a vector, but velocity (how fast and in which direction an object is going) does qualify as a vector quantity. Another example of a vector quantity is force, which describes how strongly and in what direction something is being pushed or pulled. But temperature, which has magnitude but no direction, is not a vector quantity. The word “vector” comes from the Latin vehere meaning “to carry;” it was first used by eighteenth-century astronomers investigating the mechanism by which a planet is “carried” around the Sun. 1 In text, the vector nature of an object is often indicated by placing a small arrow over the variable representing the object (such as F ), or by using a bold font (such as F ), or by underlining (such as F or F ∼ ). When you begin hand-writing equations involving Vectors, it’s very important that you get into the habit of denoting Vectors using one of these techniques (or another one of your choosing). The important thing is not how you denote Vectors, it’s that you don’t simply write them the same way you write non-vector quantities. A vector is most commonly depicted graphically as a directed line seg-ment or an arrow, as shown in Figure 1.1 (a). And as you’ll see later in this section, a vector may also be represented by an ordered set of N numbers, 1 The Oxford English Dictionary . 2nd ed. 1989. 1 2 Vectors (b) (a) Figure 1.1 Graphical depiction of a vector (a) and a vector field (b). where N is the number of dimensions in the space in which the vector resides. Of course, the true value of a vector comes from knowing what it represents.
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 630 CHAPTER 9 ■ Vectors in Two and Three Dimensions 9.1 Vectors IN TWO DIMENSIONS ■ Geometric Description of Vectors ■ Vectors in the Coordinate Plane ■ Using Vectors to Model Velocity and Force In applications of mathematics, certain quantities are determined completely by their magnitude—for example, length, mass, area, temperature, and energy. We speak of a length of 5 m or a mass of 3 kg; only one number is needed to describe each of these quantities. Such a quantity is called a scalar. On the other hand, to describe the displacement of an object, two numbers are re- quired: the magnitude and the direction of the displacement. To describe the velocity of a moving object, we must specify both the speed and the direction of travel. Quantities such as displacement, velocity, acceleration, and force that involve magnitude as well as direction are called directed quantities. One way to represent such quantities math- ematically is through the use of Vectors. ■ Geometric Description of Vectors A vector in the plane is a line segment with an assigned direction. We sketch a vector as shown in Figure 1 with an arrow to specify the direction. We denote this vector by AB > . Point A is the initial point, and B is the terminal point of the vector AB > . The length of the line segment AB is called the magnitude or length of the vector and is denoted by 0 AB > 0 . We use boldface letters to denote Vectors. Thus we write u AB > . Two Vectors are considered equal if they have equal magnitude and the same direction.
eBook - PDF- Daniel Kleppner, Robert Kolenkow(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
Rather than interrupt the flow of discussion later, we are taking time now to ensure they are on hand when required. 1.2 Vectors The topic of Vectors provides a natural introduction to the role of math-ematics in physics. By using vector notation, physical laws can often be written in compact and simple form. Modern vector notation was invented by a physicist, Willard Gibbs of Yale University, primarily to simplify the appearance of equations. For example, here is how New-ton’s second law appears in nineteenth century notation: F x = ma x F y = ma y F z = ma z . In vector notation, one simply writes F = m a , where the bold face symbols F and a stand for Vectors. Our principal motivation for introducing Vectors is to simplify the form of equations. However, as we shall see in Chapter 14 , Vectors have a much deeper significance. Vectors are closely related to the fundamen-tal ideas of symmetry and their use can lead to valuable insights into the possible forms of unknown laws. 1.2.1 Definition of a Vector Mathematicians think of a vector as a set of numbers accompanied by rules for how they change when the coordinate system is changed. For our purposes, a down to earth geometric definition will do: we can think of a vector as a directed line segment . We can represent a vector graphi-cally by an arrow, showing both its scale length and its direction. Vectors are sometimes labeled by letters capped by an arrow, for instance A , but we shall use the convention that a bold face letter, such as A , stands for a vector. To describe a vector we must specify both its length and its direction. Unless indicated otherwise, we shall assume that parallel translation does not change a vector. Thus the arrows in the sketch all represent the same vector. 1.3 THE ALGEBRA OF Vectors 3 If two Vectors have the same length and the same direction they are equal.
eBook - PDFLearning and Teaching Mathematics using Simulations
Plus 2000 Examples from Physics
- Dieter Röss(Author)
- 2011(Publication Date)
- De Gruyter(Publisher)
In quantum mechanics one works with Vectors in the infinitely dimensional Hilbert space. Plane problems can be described by two-dimensional Vectors that can be considered to lie in the complex plane. Vector algebra and vector analysis, in which partial differentiations take place, are an especially important mathematical tool of theoretical physics and therefore are often treated in depth in many textbooks for first year students.Their objects and oper-ations are not easily accessible to the untrained imagination. Therefore, the following sections concentrate only on the interactive visualization of fundamental aspects. 8.2 3D-visualization of Vectors The classical visual presentation of a vector is an arrow in space, whose length defines an absolute value and whose orientation defines a direction. The place at which the arrow is situated is arbitrary; one can, for example, let it start as a zero-point vector from the origin of a Cartesian system of coordinates. Thus its endpoint (the tip of the arrow) is described by the three space coordinates x; y; z in this system of coordinates. Its length a , also referred to as the absolute value of the vector, is obtained from the theorem of Pythagoras as a D p x 2 C y 2 C z 2 . It obviously does not matter how the system of coordinates, with respect to which the coordinates of the vector are defined, is orientated in space. Under a change of the coordinate system (translation or rotation), the individual coordinates also change, but the position and length of the vector are not affected by this. They are invariant under translation and rotation. This property provides the definition of a vector. Quantities that can be characterized by specifying a single number for every point in space are called scalar , in contract to Vectors; an example would be a density-or temperature distribution. The three-dimensional zero-point vector represents the position coordinates of a point in space.
eBook - PDF- William Cox(Author)
- 1998(Publication Date)
- Butterworth-Heinemann(Publisher)
8.1 Introduction: what is a vector? You are probably aware from your previous mathematics that one of the most powerful tools for dealing with functions of more than one variable is the use of vector notation. Thus, when describing the motion of a projectile in two dimen-sions we have the option of using (x(t), yet)) coordinates to represent position at any time or of using the position vector ret) = x(t)i + y(t)j with i, j the usual basis Vectors. The point about using Vectors here is that it effec-tively reduces the number of 'variables' -instead of two coordinates, we have one vector. A natural question therefore is to what extent one can use Vectors in the theory of functions of several variables -which is what this book is about. So far, we have managed virtually without them -and indeed it is not always clear that they would be of much use. But in fact they are -the bulk of the rest of this book is essentially about 'vector calculus'. Why should this be so? Why does most of the differen-tiation and integration that we cover occur in a vector context? Is it just the case that Vectors are simply a good shorthand notation whenever we have more than one variable? No, there is a deep significance to the idea of a vector which is simply not brought out in the introductory treatments to which you may have been exposed so far. Vectors are of fundamental importance and utility in all physical applications -they are far more than simply a shorthand notation. The reason for this is rather subtle, and requires a leap into abstraction which really has to be deferred until sufficient mathematical foundations have been laid -that is why elementary treatments are usually incomplete. To get an insight into the new view-point, let us stick with our two-dimensional projectile. You may have a number of different definitions, but they are most likely to come from the following list: 1. Any quantity having both a magnitude and a direction, such as velocity as opposed to speed.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
2 | Vectors Figure 2.1 A signpost gives information about distances and directions to towns or to other locations relative to the location of the signpost. Distance is a scalar quantity. Knowing the distance alone is not enough to get to the town; we must also know the direction from the signpost to the town. The direction, together with the distance, is a vector quantity commonly called the displacement vector. A signpost, therefore, gives information about displacement Vectors from the signpost to towns. (credit: modification of work by “studio tdes”/Flickr) Chapter Outline 2.1 Scalars and Vectors 2.2 Coordinate Systems and Components of a Vector 2.3 Algebra of Vectors 2.4 Products of Vectors Introduction Vectors are essential to physics and engineering. Many fundamental physical quantities are Vectors, including displacement, velocity, force, and electric and magnetic vector fields. Scalar products of Vectors define other fundamental scalar physical quantities, such as energy. Vector products of Vectors define still other fundamental vector physical quantities, such as torque and angular momentum. In other words, Vectors are a component part of physics in much the same way as sentences are a component part of literature. In introductory physics, Vectors are Euclidean quantities that have geometric representations as arrows in one dimension (in a line), in two dimensions (in a plane), or in three dimensions (in space). They can be added, subtracted, or multiplied. In this chapter, we explore elements of vector algebra for applications in mechanics and in electricity and magnetism. 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.
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- Henry Ricardo(Author)
- 2009(Publication Date)
- Chapman and Hall/CRC(Publisher)
. . x n 2 6 6 6 4 3 7 7 7 5 , then scalar multiplication of the vector x by the number k is de fi ned by k x ¼ kx 1 kx 2 . . . kx n 2 6 6 6 4 3 7 7 7 5 . The number k in this case is called a scalar (or scalar quantity ) to distinguish it from a vector. Because a scalar multiple of a vector in R n is again a vector in R n , we say that R n is closed under scalar multiplication . 6 A Modern Introduction to Linear Algebra 1.1.4 Geometric Vectors in R 2 and R 3 Even though the concept of a vector is simple, Vectors are very important in scienti fi c applications. To a physicist or engineer, a vector is a quantity that has both magnitude (size) and direction , for example, velocity, acceleration, and other forces. In R 2 , we can view Vectors themselves, vector addition, and scalar multiplication in a nice geometric way that is consistent with the physicist ’ s view. A vector x y ! is interpreted as a directed line segment, or arrow, from the origin to the point ( x , y ) in the usual Cartesian coordinate plane (Figure 1.1):* The magnitude of a vector, a nonnegative quantity, is indicated by the length of the arrow. The direction of such a geometric vector is deter-mined by the angle u which the arrow makes with the positive x -axis (measured in a counterclockwise direction). The addition of Vectors is carried out according to the Parallelogram Law , and the sum of two Vectors is usually referred to as their resultant (vector) (Figure 1.2). Multiplication of a vector by a positive scalar k does not change the direction of the vector, but affects its magnitude by a factor of k . Multi-plication of a vector by a negative scalar reverses the vector ’ s direction and affects its magnitude by a factor of j k j (Figure 1.3). Similarly, in R 3 , a vector x y z 2 4 3 5 can be interpreted as an arrow connect-ing the origin (0, 0, 0) to a point ( x , y , z ) in the usual x – y – z plane.
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 - PDFElectromagnetics and Transmission Lines
Essentials for Electrical Engineering
- Robert Alan Strangeway, Steven Sean Holland, James Elwood Richie(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
1 Vectors, Vector Algebra, and Coordinate Systems CHAPTER MENU 1.1 Vectors, 1 1.2 Vector Algebra, 4 1.2.1 Dot Product, 4 1.2.2 Cross Product, 7 1.3 Field Vectors, 10 1.4 Cylindrical Coordinate System, Vectors, and Conversions, 12 1.4.1 Cartesian (Rectangular) Coordinate System: Review, 12 1.4.2 Cylindrical Coordinate System, 13 1.5 Spherical Coordinate System, Vectors, and Conversions, 19 1.6 Summary of Coordinate Systems and Vectors, 25 1.7 Homework, 27 Motivation Why spend a chapter on Vectors? The degree to which you master the vector concepts and techniques in this chapter will generally determine the degree to which you understand and can apply the essential concepts and techniques of electromagnetic fields. Vectors are not just important tools in the calculation of electromagnetic field quantities. They are essential in the visualization of electromagnetic fields in an organized, dependable man- ner. In short, a mastery of Vectors instills the “thought infrastructure” that allows one to visualize and apply elec- tromagnetic fields in electrical engineering applications. Enjoy! 1.1 Vectors This chapter develops the tools necessary to define and manipulate various types of Vectors in three different coor- dinate systems. We begin with how Vectors can be used to describe location and displacement. What is a vector? It is a quantity with magnitude (scalar) and direction. What are examples of Vectors? Velocity, force, acceleration, and so forth. How is direction in three-dimensions expressed? Start with the following vector definition: A position vector r locates a position in space with respect to the origin, that is, it is a vector that starts at the origin and ends at the point of the designated position in space (notation r is also used in some textbooks and literature). 1 Electromagnetics and Transmission Lines: Essentials for Electrical Engineering, Second Edition. Robert A. Strangeway, Steven S. Holland, and James E.
- Robert E. White(Author)
- 2006(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 2 Vectors in Space Vectors in space are introduced, and the dot, cross and box products are stud-ied. Lines and planes are carefully described as well as extensions to higher dimensional space. Applications to work, torque, inventories and visualizations are included. 2.1 Vectors and Dot Product A point in space can be located in a number of ways, but here the Cartesian coordinate system will be used. You may wish to visualize this from the interior of a room looking down into the corner. The corner is the origin; the x-axis is the intersection of the left wall and floor; the y-axis is the intersection of the right wall and the floor; the intersection of the left and right walls is the z-axis. This is illustrated in Figure 2.1.1. The point ( ) is located by moving units in the x-axis, then moving units parallel to the y-axis, and moving units parallel to the z-axis. The distance from the origin to the point is given by two applications of the Pythagorean theorem to the right triangles in Figure 2.1.1. Associated with the point ( ) is the position vector from the origin to this point. Definition 2.1.1. A vector in R 3 is an ordered list of three real numbers = [ 1 2 3 ] . One can visualize this by forming the directed line segment from the origin point (0 0 0) to the point ( 1 2 3 ) Notation. Points in R 3 will be denoted by ( 1 2 3 ) and Vectors will be represented by either row or column Vectors: = [ 1 2 3 ] indicates a row vector a = 1 2 3 indicates a column vector a = [ 1 2 3 ] is called the transpose of the column vector a so that a = 47 48 CHAPTER 2. Vectors IN SPACE Figure 2.1.1: Point in Space Example 2.1.1.
eBook - PDFLinear Algebra
A First Course with Applications
- Larry E. Knop(Author)
- 2008(Publication Date)
- Chapman and Hall/CRC(Publisher)
Two points determine a line segment, and if we orient the space properly then we can put the line segment where we want. As usual we have both position vector and free vector representations. v v v v v Free vector representation Position vector representation v 0 0 0 x 1 x 2 … … x n FIGURE 4 Example 3: As a numerical illustration of the geometry, suppose one representation of the vector v in R 4 starts at the point P (2,7,0, 4) and ends at the point Q (5,7, 2,1). Then we have v ¼ 5 2 7 7 2 0 1 ( 4) 2 6 6 4 3 7 7 5 ¼ 3 0 2 5 2 6 6 4 3 7 7 5 . If the same vector v started at (4,4,8, 2) then it would end at the point (4 þ 3,4 þ 0,8 þ ( 2), ( 2) þ 5), which is (7,4,6,3). If the same vector v ended at the point (1,2,6,5) then v would start at the point (1 3,2 0,6 ( 2), 5 5), which is ( 2,2,8,0). As with R 2 , we can give geometric meaning to the sum of two Vectors in R n and to the scalar multiple of a vector by a number. Consider R 3 fi rst, because we can draw reasonable An Introduction to Vector Spaces & 97 pictures of R 3 on paper. For two Vectors u and v in R 3 , all aspects of the sum are determined by three points: the origin, the tip of u , and the tip of v . Three points determine a plane. Thus the Vectors u and v , and their sum, are embedded in R 3 , but all the Vectors lie in a single plane and if we restrict our attention to that plane then the picture is the picture we drew for R 2 . In particular, the sum u þ v is just the diagonal of the parallelogram determined by u and v . The picture of a vector u multiplied by a number c is even simpler because, while we are in R 3 , the only points that matter are the origin and the tip of u , so we may restrict our vision to a line. Thus the picture of c u in R 3 looks very much like the picture we drew of a scalar multiple in R 2 , or like one in R 1 for that matter.
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