FLEXIBLE LEARNING APPROACH TO PHYSICS Module M2.4 Introducing scalars and vectors 1 Opening items 3 Introducing vector algebra 1.1 Module introduction 3.1 Scaling vectors 1.2 Fast track questions 3.2 Adding and subtracting vectors 1.3 Ready to study? 3.3 Resolving vectors 2 Scalars and vectors 4 Working with vectors 2.1 Scalars and scalar quantities 5 Closing items 2.2 Vectors and vector quantities 5.1 Module summary 2.3 Representing scalars and vectors 5.2 Achievements 5.3 Exit test Exit module FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 1 Opening items 1.1 Module introduction Many of the quantities that physicists use are easy to specify; measurements of mass, length, time, area, volume and temperature can all be expressed as simple numbers together with appropriate units of measurement. Such quantities are known as scalar quantities or scalars. However, some quantities are more tricky to deal with. For instance, if you want to travel from one place to another you will not only want to know how far apart the two places are, you will also need to know the direction that leads from one to the other. The physical quantity that combines distance and direction is called displacement and is clearly more complicated than distance alone. Displacement is a simple example of a large class of physical quantities known collectively as vector quantities or vectors. A detailed understanding of vectors and how to use them is crucial to many parts of physics, particularly mechanics and electromagnetism. FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 This module provides an introduction to the mathematical treatment of vectors. Section 2 gives a definition of vector quantities that separates them from simpler scalar quantities1—1such as mass and distance1—1that are also defined. This section also describes the graphical representation of vectors and the notation used to distinguish vectors from scalars, both in print and in handwritten work. Section 3 introduces some of the basic operations of vector algebra such as scaling (multiplication by a scalar), vector addition and vector subtraction. It aims to make clear the meaning of vector equations such as 3a2−22b2=20. The section ends with a discussion of the way in which a given vector may be split-up (resolved1) into component vectors1—1a process that is of importance in many practical problems. Throughout the module the emphasis is on basic ideas and geometric (graphical) methods. The algebraic methods that are more frequently used by those already familiar with vectors are mentioned briefly in a short conclusion (Section 4) but their full development is left to other modules. Since it is those methods that are best suited to tackling three-dimensional problems, most of the questions in this module are restricted to two- dimensional situations on idealized planes or flat surfaces. However, even though the discussion is mainly restricted to two dimensions, the significance of vectors is amply demonstrated as is the importance of always using vector notation clearly and correctly. Study comment Having read the introduction you may feel that you are already familiar with the material covered by this module and that you do not need to study it. If so, try the Fast track questions given in Subsection 1.2. If not, proceed directly to Ready to study? in Subsection 1.3. FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 1.2 Fast track questions Study comment Can you answer the following Fast track questions? If you answer the questions successfully you need only glance through the module before looking at the Module summary (Subsection 5.1) and the Achievements listed in Subsection 5.2. If you are sure that you can meet each of these achievements, try the Exit test in Subsection 5.3. If you have difficulty with only one or two of the questions you should follow the guidance given in the answers and read the relevant parts of the module. However, if you have difficulty with more than two of the Exit questions you are strongly advised to study the whole module. Question F1 In terms of a strict interpretation of vector and scalar quantities, what is wrong with each of the following statements? (a) The velocity of light travelling through a vacuum (usually denoted by c) is approximately 32×21081m1s1−11. (b) The acceleration due to gravity (usually denoted by g) near the surface of the Earth is approximately 101m1s1−12. (c) The mass of your head pushes down on your neck. FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 Question F2 a is a displacement of 4.321km due north, and b is a displacement of 2.401km due east. (a) What is the magnitude and the direction of the displacement 3a2−25b? (b) Given a specified line, any vector may be expressed as the sum of two (orthogonal) component vectors, one parallel to the given line and the other at right angles to that line. What is the magnitude of the (orthogonal) component vector of 3a2−25b that points in the direction 20° west of north? Study comment Having seen the Fast track questions you may feel that it would be wiser to follow the normal route through the module and to proceed directly to Ready to study? in Subsection 1.3. Alternatively, you may still be sufficiently comfortable with the material covered by the module to proceed directly to the Closing items. FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 1.3 Ready to study? Study comment Relatively little background knowledge is required to study this module. Terms such as mass, electric charge and temperature are used and it is assumed that you know something about measuring such quantities using SI units, but you do not need to know precise definitions or exact meanings. It is also assumed that you have met Newton’s laws of motion before, but again you do not need to know those laws in detail in order to study the module. (Indeed, this module would be good preparation for a thorough study of Newton’s laws.) There are some mathematical topics, however, with which you will need to be familiar before starting to study; they are the subject of the following questions. Consult the Glossary for references to any terms that are unfamiliar. FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 Question R1 Draw a set of two-dimensional Cartesian axes. Label the horizontal axis x and the vertical axis y. Using the axes you have drawn, plot the three points A, B and C the coordinates of which are given in Table 1. Draw the right- angled triangle that has the points A, B and C at its corners. Write down the lengths of sides AB and BC of that triangle, and use Pythagoras’s theorem to work out the length of the side CA. Table 1 See Question R1. point x-coordinate y-coordinate A1 1 B−31 C−34 FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 Question R2 Using the triangle given in the answer to Question R1 (Figure 15), and indicating the length of side CA by c and the angle CABÃ by θ, write down expressions for the lengths of sides AB and BC in terms of the length c, the angle θ, and the basic trigonometric ratios. What is the value (in degrees) of the angle θ1? Question R3 The modulus of a quantity x is written |1x1| and represents the absolute value of x. Evaluate the following expressions and complete the equations. (a)3|131| =33(b)3|1−131| =33(c)3|1(−12)(3.1)1| =33(d)31|1(−12.4)1/1(2)1| = FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 2 Scalars and vectors 2.1 Scalars and scalar quantities Many important physical quantities, such as length, mass and temperature, can be completely specified by a single number together with an appropriate unit of measurement. For instance, it makes perfectly good sense to say that the length of an object is 1.421m or that the mass of an object is 12.21kg. Quantities that can be specified in this simple and straightforward way are called scalar quantities. Thus: Scalar quantities are physical quantities that can be completely specified by a single number together with an appropriate unit of measurement. FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 Different values of a given scalar quantity may be easily added, subtracted, multiplied or divided. In fact, all the elementary operations of arithmetic apply to values of a scalar quantity just as they do to ordinary numbers. So, if a, b and c are three values of a given scalar quantity (three masses, say) then we know that a + b = b + a (1) ab = ba (2) a + 0 = a (3) a × 1= a (4) (a + b) + c = a + (b + c) (5) ☞ a(bc)=(ab)c (6) a(b + c)=ab + ac (7) Scalar quantities are often referred to simply as scalars. There is nothing very surprising about the behaviour of scalars; their properties seem obvious and, indeed, they are obvious. What is much more interesting is that there are many physical quantities that are not scalars. FLAP M2.4 Introducing scalars and vectors COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 2.2 Vectors and vector quantities Edinburgh Figure 1 shows the relative locations of Belfast, Edinburgh and Exeter, three of the major cities in the UK.