Technology & Engineering
Cartesian Vector
A Cartesian vector is a mathematical representation of a quantity that has both magnitude and direction. It is typically represented by an ordered set of components in a coordinate system, such as x, y, and z in three-dimensional space. Cartesian vectors are widely used in engineering and technology for describing forces, velocities, and other physical quantities in a precise and systematic manner.
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6 Key excerpts on "Cartesian Vector"
- Kuldip S. Rattan, Nathan W. Klingbeil, Craig M. Baudendistel(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Two-Dimensional Vectors in Engineering CHAPTER 4 The applications of two-dimensional vectors in engineering are introduced in this chapter. Vectors play a very important role in engineering. The quantities such as displacement (position), velocity, acceleration, forces, electric and magnetic fields, and momentum have not only a magnitude but also a direction associated with them. To describe the displacement of an object from its initial point, both the distance and direction are needed. A vector is a convenient way to represent both magnitude and direction and can be described in either a Cartesian or a polar coordinate system (rectangular or polar forms). For example, an automobile traveling north at 65 mph can be represented by a two-dimensional vector in polar coordinates with a magnitude (speed) of 65 mph and a direction along the positive y-axis. It can also be represented by a vector in Cartesian coordinates with an x-component of zero and a y-component of 65 mph. The tip of the one-link and two-link planar robots introduced in Chapter 3 will be represented in this chapter using vectors both in Cartesian and polar coordinates. The concepts of unit vectors, magnitude, and direction of a vector will be introduced. 4.1 INTRODUCTION Graphically, a vector −− → OP or simply P with the initial point O and the final point P can be drawn as shown in Fig. 4.1. The magnitude of the vector is the distance between points O and P (magnitude = P) and the direction is given by the direction of the y Magnitude = P P x O θ Figure 4.1 A representation of a vector. 107 108 Chapter 4 Two-Dimensional Vectors in Engineering arrow or the angle in the counterclockwise direction from the positive x-axis as shown in Fig. 4.1. The arrow above P indicates that P is a vector. In many engineering books, the vectors are also written as a boldface P.
eBook - PDF- Stephen Lee(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
17 Use of vectors This grand book – the Universe… is written in the language of mathematics Galileo Galilei (1623) Vectors are important in modelling and solving two- and three-dimensional problems in mechanics. You have already used vector methods. This chapter will help you to review and consolidate your knowledge of vectors with particular reference to their application in mechanics. 17.1 Vector basics A vector is a quantity with both magnitude (also known as length, size or modulus) and direction. In mechanics, vectors are used to represent quantities such as displacement, velocity, acceleration, force and momentum. A vector is usually written PQ ⎯→ or a and is typeset as PQ ⎯→ or as bold PQ or a . In figure 17.1, PQ ⎯→ QA ⎯→ because their lengths and directions are the same. Figure 17.1 Vectors can be added: a b c and subtracted: a b d and multiplied by a scalar: a has magnitude magnitude of a . Figure 17.2 The length of the vector a is denoted a or a . When a 1, a is a unit vector . A unit vector is denoted by having a hat a . The position vector of a point A relative to a point O is the vector OA ⎯→ . Normally, O is the origin of Cartesian axes and the position vector of a point is usually denoted by the corresponding lower-case letter, thus OA ⎯→ a , OB ⎯→ b , and so on. Then AB ⎯→ b a , a d c 2 a a a b b P Q Q A a a 378 AN INTRODUCTION TO MATHEMATICS FOR ENGINEERS : MECHANICS where b and a are the position vectors of A and B with respect to an assumed origin (figure 17.3) which is not necessarily drawn on the diagram. AB ⎯→ OA ⎯→ OB ⎯→ OA ⎯→ OB ⎯→ a b b a Figure 17.3 Cartesian components Vectors can also be expressed in Cartesian component form, with respect to some appropriate origin and axes. In three dimensions: OA ⎯→ x i y i z k where a , y , z are the displacements represented by the vector in the x , y and z directions and i , j and k are the unit vectors in the directions of the co-ordinate axes.
eBook - PDF- Daniel A. Fleisch(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
A tensor of rank 3 may be represented by a three-dimensional array of values. With these basic definitions in hand, you’re ready to begin considering the ways in which vectors can be put to use. Among the most useful of all vectors are the Cartesian unit vectors, which you can read about in the next section. 1.2 Cartesian unit vectors If you hope to use vectors to solve problems, it’s essential that you learn how to handle situations involving more than one vector. The first step in that process is to understand the meaning of special vectors called “unit vectors” that often 6 Vectors 1 2 y 1 2 z 1 2 x i j k Figure 1.3 Unit vectors in 3-D Cartesian coordinates. serve as markers for various directions of interest (unit vectors may also be called “versors”). The first unit vectors you’re likely to encounter are the unit vectors ˆ x , ˆ y , ˆ z (also called ˆ ı , ˆ j , ˆ k ) that point in the direction of the x -, y -, and z -axes of the three-dimensional Cartesian coordinate system, as shown in Figure 1.3 . These vectors are called unit vectors because their length (or magnitude) is always exactly equal to unity, which is another name for “one.” One what? One of whatever units you’re using for that axis. You should note that the Cartesian unit vectors ˆ ı , ˆ j , ˆ k can be drawn at any location, not just at the origin of the coordinate system. This is illustrated in Figure 1.4 . As long as you draw a vector of unit length pointing in the same direction as the direction of the (increasing) x -axis, you’ve drawn the ˆ ı unit vector. So the Cartesian unit vectors show you the directions of the x , y , and z axes, not the location of the origin. As you’ll see in Chapter 2 , unit vectors can be extremely helpful when doing certain operations such as specifying the portion of a given vector pointing in a certain direction. That’s because unit vectors don’t have their own magnitude to throw into the mix (actually, they do have their own magnitude, but it is always one).
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.
eBook - PDFCalculus
Multivariable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
11.2 Vectors 663 56. Writing Explain how you might determine whether a set of points in 3-space is the graph of an equation involving at most two of the variables x, y, and z. 57. Writing Discuss what happens geometrically when equations in x, y, and z are replaced by inequalities. For example, compare the graph of x 2 + y 2 + z 2 = 1 with the set of points that satisfy the inequality x 2 + y 2 + z 2 ≤ 1. 11.1 | Quick Check Answers 1. √ 38 2. sphere; 4; (3, 2, −1) 3. √ 38 − 6 4. a. (0, 0, −3); 5 b. 4; −4 c. 2; −8 11.2 Vectors Many physical quantities such as area, length, mass, and temperature are completely described once the magnitude of the quantity is given. Other physical quantities, called “vectors,” are not completely determined until both a magnitude and a direction are specified. For example, winds are usually described by giving their speed and direction, say 20 mi / h northeast. The wind speed and wind direction together form a vector quantity called the wind velocity. In this section we will develop the basic mathematical properties of vectors. Vectors in Physics and Engineering If a particle moves along a number line, then a change in its position can be described by a signed real number. For example, a displacement of −3 describes a position change of 3 units in the negative direction along the line. However, for a particle that moves in two dimensions or in three dimensions, a signed real number is no longer sufficient to specify displacement—other methods are required. One method is to use an arrow, called a vector, drawn from the starting point to the ending point; this is called the displacement vector from the first point to the second. For example, Figure 11.2.1a shows the displacement vector of a particle that moves from point A to point B along a circuitous path. Note that the length of the arrow describes the distance between the starting and ending points and not the actual distance traveled by the particle.
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.
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