VECTORS & SCALARS (A. Savas ARAPO GLU) June 17, 2018 Contents 1 Introduction 2 2 Units and Dimensions 2 2.1 Dimension and Dimensional Analysis...................... 3 3 Scalars & Vectors 3 3.1 Scalars....................................... 3 3.2 Vectors....................................... 4 3.3 Coordinate-Free Properties of Vectors...................... 4 4 Coordinate Sytems & Unit Vectors 5 4.1 Unit Vectors.................................... 6 5 Multiplication of Vectors 7 5.1 Scalar Product of Vectors............................ 7 5.2 Vector Product of Vectors............................ 8 1
1 Introduction Before considering the laws of motion, we need a considerable amount of preparatory training even to learn what the words mean, e.g. vectors, scalars etc. This chapter is all about the denitions of mathematical structures that will be used later to express the laws of motion in a systematic way... (Read the sections 1.1,2) 2 Units and Dimensions The principle of science is simply stated as the test of all knowledge is experiment and the experiment is the sole judge of scientic/physical truth. Thus, Science & Physics rest on experiment and observation, and experiment and observation require measuring some quantities. Four basic entities, or fundamental quantites of classical physics to be measured are L, T, M, and Q. In phys I (Fiz101E), all measurements reduce ultimately to the measurement of length, time, and mass. The measurement of any quantity (e.g. mass) is made relative to a particular standard, or unit. Then, how can we choose these standards/units? Actually there are many dierent choices but the most commons are: MKS(SI) CGS L m cm T s s M kg g where SI is mostly for engineering applications, and CGS for theoretical physics. Figure 1: 7 base units of SI The units shown in the Fig. (1) are fundamental, or base units; there are also derived units, e.g. the force: In SI, the unit of force is kg.m/s 2 N and in CGS g.cm/s 2 dyne. 2
2.1 Dimension and Dimensional Analysis The dimension of a quantity is the `type' of base quantites that constitute the given quantity; for example, the (physical) dimension of velocity is L/T - then knowing this, you can determine the unit of velocity depending on your choice. Any physical quantity, no matter how complex, can be expressed as an algebraic combination of these 3 basic quantities in Phys I. We can express the relation between the units and dimension in the following way: A unit is the scale with which a dimension is measured.. But, DO NOT CON- FUSE THE DIMENSION OF A QUANTITY WITH THE UNITS IT IS MEASURED! The dimensional anlysis is simply to check the both sides of an equation whether they are dimensionally consistent - this is helpful in nding errors in an equation (but do not give us any information about the constant factors in the equations). For example, for the period of a simple pendulum, is it T = 2π g or T = 2π l? l g Another important notion about measurements is how to report the result of measurements, i.e. how to write numbers? The closely related concepts are precision, accuracy; signicant gures; uncertainty in lab classes... Example: What is the (physical) dimension of the force? Example: Assume that a simple formula for the speed of water waves is in the form v = kg a h 1 λ b, where v is the speed k is a dimensionless number, g is the gravitational acceleration, h is the depth of water, λ is the wavelength, and a and b are two constants. Find a and b. (Read the sections 1.3,4,6) 3 Scalars & Vectors The concept of the observer (people doing experiment or observation) is an indispensible part of science. In this course, whenever we say, for example, the position of an object, we implicitly mean the position in a specic coordinate system (or a coordinate frame). Actually the coordinate system is a fancy word for the observer in physics. Thus, to analyze the motion of an object in space, we must rst choose a coordinate system (that is, there must be an observer). Scalars and vectors are mathematical quantities which are used in expressing the laws of motion; so they must be dened through their relation to the observers (or coordinate systems). 3.1 Scalars A scalar is dened to be any quantity whose value does not depend on coordinate systems; e.g. the (rest) mass of an object. Practically a scalar is an algebraic quantity with a proper unit and without an associated direction; this, in turn, implies that 3
a scalar is just a number but the reverse is not always true! 3.2 Vectors The word vector means carrier in Latin; biologists use the term vector to mean an insect, animal, or an other agent that carries a cause of a disease from one organism to another. The denition of the term vector in physics (at the level of 101) is: Vectors are the quantities that have both a magnitude and a direction and that must follow certain rules of addition - Parallelogram rule!. Thus, practically, a vector is specied more than just a single number. All quantities having both a magnitude and a direction are not necessarily vectors; for example, nite angle rotations cannot be represented by vectors although they have a direction and magnitude; see the Fig. (2): Figure 2: Finite angle rotations are not represented by vectors because their addition does not satisfy the commutativity of the addition. Notation: A denotes the vector A; A = A = A denotes the magnitude/size/length of A. Note that the magnitude of a vector is a non-negative number and it is coordinateindependent, thus it a scalar. 3.3 Coordinate-Free Properties of Vectors This is basically the high school physics (lise zi gi)... A = B both the magnitudes and the directions are the same. 4
Vector Addition: Since a vector has both a magnitude and a direction, vector addition does not obey the rules of ordinary algebra. We must dene a procedure for adding vectors!. The method dened is the so-called Head-to-tail method, or parallelogram method, shown in Fig. (3) Figure 3: Head-to-tail method, or parallelogram method. Vector addition is commutative ( A + B = B + A) and associative (( A + B) + C = A + ( B + C)). Multiplication by a number: 2 A, A... Subtraction... Importance of Vector Notation: Vector notation is concise. (Many physical laws have simple and transparent forms which are disguised when these laws are written in a particular coordinate system.) When you express a law of physics in the form of a vector equation, it holds in all coordinate systems (form invariance); for example, F = m a and F = m a... 4 Coordinate Sytems & Unit Vectors In solving a problem, we choose a coordinate system which (should) simplify the solution. Thus, we must develop a language to describe a vector in a coordinate system. But in this course we will consider only the cartesian (rectangular) coordinate systems. (You know that there is also polar coordinates r and θ in the 2-dimensional Euclidean plane. Actually there are many types of coordinate systems that can be chosen depending on the symmetries of the system under consideration). 5
Figure 4: Cartesian components of a vector A The cartesian components of the vector A are expressed as: A x = Acosθ, (1) A y = Asinθ. (2) where A is the magnitude of the vector A. We must note that these are the components of A in this coordinate system. (That is, we can choose another coordinate system in which the components can be dierent although the vector is the same. This means that the components of a vector are coordinate-dependent - they are numbers but they are not scalars!) The magnitude of the vector is: A = A 2 x + A 2 y, and the direction of the vector refers to the quantity tanθ = Ay A x which is the angle the vector makes with the +x-axis in the counterclockwise direction. 4.1 Unit Vectors A unit vector is a vector of unit magnitude; they only specify the direction and they are dimensionless (and unitless), so they are just mathematical devices. In cartesian coordinate systems the unit vectors being used are denoted by ˆx, ŷ, and ẑ (or î, ĵ, and ˆk); being a unit vector, they satisfy ˆx = ŷ = ẑ = 1. The vector A in the gure above is written in terms of them as A = A xˆx + A y ŷ. Vector operations are especially simple with components and unit vectors because you do not need to visualize vectors explicitly in a coordinate system to add or subtract them, for example, A ± B = (A x ± B x )ˆx +, c A = ca xˆx +. Example: Is the vector î + ĵ a unit vector? Given a vector A = 2î 3ĵ, construct a unit vector in the same direction as A. Draw these vectors on a two-dimensional coordinate system. (Read the sections 1.7,8,9. You can solve the suggested problems 1.49,51.) 6
5 Multiplication of Vectors 5.1 Scalar Product of Vectors The scalar product or dot product is dened as A B ABcosθ, (3) where θ is the angle between the vectors for the conguration shown in the gure below: Figure 5: Scalar product of two vectors and the angle between them. From this denition, we can reach the followings: The result of this product is coordinate-independent, so the result is a genuine scalar - that is why this product is called as scalar product. A B = B AB. A ( B + C) = A B + A C. If θ = π/2 rad, then A B = 0. A A = A 2 A = A A. We can express the scalar product in terms of the cartesian components of the vectors. Using the followings: (Note that A A = A 2 x + A 2 y + A 2 z.) î î = ĵ ĵ = ˆk ˆk = 1 (4) î ĵ = î ˆk = ĵ ˆk = 0, (5) A B = A x B x + A y B y + A z B z. (6) 7
We could have dened A B to be any operation at all, for example, A B =..., but this would turn out to be of no use to us in physics. With this denition, a number of physical quantities can be described as the scalar product of two vectors, e.g. work. Example: (a) Given two vectors A = 2î 3ĵ and B = î + 3ĵ, calculate A B. (b) If the two vectors A = nî ĵ and B = î + 3ĵ are perpendicular to each other, nd n (n is a constant). (c) Given A = 2î 3ĵ + ˆk and B = î 3ˆk, nd the angle between them; nd the angle between A and the y-axis. 5.2 Vector Product of Vectors In dening the vector product or cross product, we should introduce two inputs: one for the magnitude and the other for the direction! where the magnitude of the C is dened as and the direction is by RHR! C = A B, (7) C = ABsinθ, (8) Figure 6: Vector product of two vectors and the RHR From this denition, we can reach the followings: C is perpendicular to both A and B. 8
RHR leads to the anti-commutativity of the vector product: A B = B A. If θ = 0 or θ = π rad, then C = 0 for non-zero vectors A and B. This also implies that the cross product of any vector with itself is zero, i.e. A A = 0. To obtain the result of cross product in terms of the cartesian components of vectors, we must rst obtain the cross product of cartesian unit vectors. For the self multiplications, it is obvious that î î = ĵ ĵ = ˆk ˆk = 0. (9) For the right-handed coordinate systems all over the world, conventionally we take î ĵ = ˆk 1, then ĵ ˆk = î and ˆk î = ĵ. Now using the rules for the cross product of the unit vectors we can express the cross product of two vectors by their cartesian coordinates, as in the case of dot products. There is, on the other hand, a systematic way of getting cross products of two vectors by using Matrix determinants: A B î ĵ ˆk = A x A y A z = î(a y B z A z B y ) ĵ() + ˆk(). (10) B x B y B z Torque, angular momentum are expressed in terms of the vector product. (Read the section 1.10. You can solve the suggested problems 1.86,89,93,95,96,102) Example: (a) Given two vectors A = 2î 3ĵ and B = î + 3ĵ, nd a third vector C which is perpendicular to both A and B. Example: (1.96) You are given two vectors A = 5.0î 6.5ĵ and B = 3.5î + 7ĵ. A third vector Ĉ lies in the xy-plane, and it is perpendicular to A and its scalar product with B is 15.0. From this information, nd the components of C. Example: The vector r = xî + yĵ + zˆk, called the position vector, points from the origin (0, 0, 0) to an arbitrary point (x, y, z). For a moving object this vector is time-dependent: r(t) = x(t)î + y(t)ĵ + z(t)ˆk, where x(t), y(t), and z(t) are called coordinate functions. Actually the whole aim of the classical physics is to nd this vector for a particle because if you know it you have the all information about the motion of the particle in a deterministic sense. In the next two chapters (kinematics) we will see how to construct this object, but now for a both a practicing with vectors and for a warm-up for the next chapter consider the following problem. (a) If an object is moving with constant velocity, its position vector in a coordinate system is given as r(t) = r 0 + vt. Assume that in a two-dimensional coordinate system an object, initially at point (1 m, 1 m), is moving with a constant velocity v = 2î + ĵ m/s. Find the 1 This choice amounts to the fact that one chooses x and y directions freely but then the z direction is xed with this conventional choice 9
functions x(t) and y(t) giving the position of the object as a function of time. (b) Find its distance from the origin at t = 2 s. (This problem is actually a warm-up for the next chapter.) (c) If an object is moving with constant acceleration, its position vector in a coordinate system is given as r(t) = r 0 + v 0 t + 1 2 at2. For projectile motion obtain the coordinate functions! 10