Vector Algebra

CHAPTER – I VECTOR ALGEBRA INTRODUCTION: Electromagnetic Field (EMF) Theory is often called Electromagnetics. It is a subject which deals with electric field, magnetic field and also electromagnetic fields and phenomena. EMF Theory is essential to analyze and design all communication and radar systems. Infact, it is also used in Bio-Systems and in this context it is called BioElectromagnetics. Source of electromagnetic field is electric charges: either at rest or in motion. However an electromagnetic field may cause a redistribution of charges that in turn change the field and hence the separation of cause and effect is not always visible. APPLICATIONS OF EMF THEORY: 1. Communication Systems. Wireless Communication Satellite Communication TV Communication Cellular Communication Mobile Communication Microwave Communication Fibre-Optic Communication. 2. Electrical Machines Electro mechanical Energy conversion systems. Electrical Motors Transformers Electrical Relays 3. Radars Speed-trap radars Weather forecast radars Remote sensing radars Radio astronomy radars Meteorological radars 4. Industries Induction heating Melting and forging Surface hardening Annealing Soldering Dielectric heating 5. All types of antenna analysis and design. 6. All types of transmission lines and wave guides. The analysis and design of a system, device or circuit requires the use of some theory or the other. 1

The analysis of a system is universally defined as one by which the output is obtained from the given input and system details. The design of a system is one by which the system details are obtained from the given input and output. These two important tasks of analysis and design are executed by two most popular theories, namely, CIRCUIT and ELECTROMAGNETIC theories. COMPARISON OF CIRCUIT AND ELECTROMAGNETIC FIELD THEORIES Sl.No. 1 2 3 4 5

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Circuit Theory Deals with Voltage (V) and Current (I). V and I are scalars. V and I are functions of time (t). Radiation effects are neglected. Circuit theory cannot be used to analyze or design a complete communication system. It is useful at low frequencies. At low frequencies the length of connecting wires is very much smaller than λ. Cannot be applied in free space. Using circuit theory, transmitter and receiver circuits can be analyzed and designed. But it cannot be used to design or analyze a medium like free space. Basic laws are Ohm’s law and Kirchhoff’s law. Basic theorems are Thevenin’s, Norton’s, Reciprocity, Superposition, Maximum power transfer theorems. Basic equations are Mesh/Loop equations.

EMF Theory Deals with Electric (E) and Magnetic (H) fields. E and H are vectors. E and H are functions of time (t) and space variables (x, y, z) or (ρ, φ, z) or (r, θ, φ). Radiation effects can be considered. Field theory can be used where circuit theory fails to hold good for the analysis and design of a communication system. It is useful at all frequencies, particularly at high frequencies. At high frequencies the length of connecting components are of the order of λ. Is applicable in free space. Using field theory, the medium can also be analyzed and designed.

Basic laws are Coulomb’s law, Gauss’s law, Ampere’s circuit law. Basic theorems are Reciprocity, Helmholtz, Stoke’s, Divergence and Poynting theorems. Basic equations are Maxwell, Poisson, Laplace and Wave.

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VECTOR ANALYSIS: The analysis and design of a system or device is impossible without mathematics. Mathematics is the backbone of science and engineering. For all analytical or computational purposes, mathematical background is essentia l. Mathematical modeling of systems is a common practice. Vector analysis is a mathematical tool with which EM concepts are most conveniently expressed and best comprehended. Vectors are introduced in physics and mathematics, primarily in the Cartesian coordinate system. Although Cylindrical coordinates are found in Calculus texts, the spherical coordinate system is seldom used. All three coordinate systems must be used in electromagnetics. A quantity can either be a Scalar or Vector. The term scalar refers to a quantity whose value may be represented by a single positive or negative real number. In other words, a scalar is a quantity that has only magnitude. The x, y, z we speak in basic algebra are scalars and the quantities they represent are scalars. Quantities such as time, mass, distance, temperature, entropy, electric potential and population are scalars. Voltage is also a scalar quantity, although the complex representation of a sinusoidal voltage, an artificial procedure, produces a complex scalar, or phasor, which requires two real numbers for its representation such as amplitude and phase angle, or real part and imaginary part. A vector quantity has both magnitude and direction. We shall be concerned with only two and three – dimensional spaces, but vectors are defined in n - dimensional space in more advanced applications. Force, velocity, acceleration, electric field intensity and a straight line from the positive to the negative terminal of a storage battery are examples of vectors. Each quantity is characterized by both magnitude and direction. There is other class of physical quantities called Tensors: where magnitude and direction vary with coordinate axes. To distinguish between a scalar and a vector, it is customary to represent a vector by a letter with an arrow on top of it, such as Ā or Ē, or by a letter in boldface type such as A and B. A scalar is represented simply by a letter – e.g. A, B, U and V. EM theory is essentially a study of some particular fields. A field may be defined mathematically as some function of a vector which connects an arbitrary origin to a general point in space. Field concept is invariably related to a region. It is possible to associate some physical effect with a field, such as the force on a compass needle in the earth’s magnetic field or the movement of smoke particles in the field defined by the vector velocity of air in some region of space. Both scalar fields and vector fields exist. The temperature throughout the bowl of a soup, the density at any point in the earth, sound intensity in a theatre, electric potential in a region and refractive index of a stratified medium are examples of scalar fields. The gravitational and magnetic fields of the earth, the voltage gradient in a cable, the temperature gradient in a soldering – iron tip and velocity of rain drops in the atmosphere are examples of vector fields. The value of a field varies in general with both position and time.

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UNIT VECTOR: A vector A has both magnitude and direction. The magnitude of A is a scalar written as A or |A|. A Unit Vector, a A along A is defined as a vector whose magnitude is unity and its direction is along A.

aA = A/|A| = Ā/A We have, |a A | = 1. Thus, A can be written as A = A aA which completely specifies A in terms of its magnitude A and its direction a A . To describe a vector in the Rectangular Coordinate System, let us first consider a vector A or Ā extending outward from the origin. This vector A can be identified by three component vectors, lying along the three coordinate axes, whose vector sum must be the given vector. If the component vectors of the vector A are x, y, and z, then, A = x + y + z. In other words, the component vectors have magnitudes which depend on the given vector, but they each have a known and constant direction. This suggests that the unit vectors have unit magnitude and directed along the coordinate axes in the direction of the increasing coordinate values. Thus, vector Ā in Cartesian coordinates may be represented as Ā = (A x, Ay , Az ) Or Ā = Ax a x + Ay a y + Az a z where A x, Ay , Az are called the components of A in the x, y and z directions respectively and a x , a y, a z are unit vectors in the x, y and z directions. The magnitude of vector Ā or A is given by A = Ā = √(Ax² + Ay ² + Az²) and the unit vector along A or Ā is given by aA = (Ax ax + Ay ay + Az az)/ √(Ax² + Ay ² + Az²) Properties of Unit Vectors: i. ax . ax = ay . ay = az . a z = 1 ii. ax x ax = ay x a y = az x a z = 0 iii. ax x ay = a z iv. ax . ay = 0 v. ay x ax = - az

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Vector Algebra: 1. Addition and Subtraction: Two vectors A and B can be added together to give another vector C, that is C=A+B The vector addition is carried out component by component. Thus, if A = (Ax , Ay, Az) and B = (Bx,By, Bz) , C = (Ax + Bx) a x + (Ay + By) a y + (Az + Bz) a z Vector subtraction is similarly carried out as D = A – B = A + (-B) = (Ax - Bx) a x + (Ay - By) a y + (Az - Bz) a z Graphical representation of Vector Addition and Subtraction:

Vector addition obeys i. Commutative law A + B = B + A (Addition) ii. Associative law A + (B + C) = (A + B) + C iii. Distributive law k(A + B) = kA + kB

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kA = A k (Multiplication)

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k(lA) = (kl)A

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Position and Distance Vector: A point P in Cartesian coordinates may be represented by (x, y, z). “The position vector, rp (or radius vector) of point P is as the directed sitance from the origin O to P.” rp = OP = x a x + y a y + z a z. The position vector of point P is useful in defining its position in space. Ex: Point P (3, 4, 5) and its position vector 3 a x + 4 a y + 5 a z are as shown. “The distance vector is the displacement from one point to another.” If two points ‘P’ and ‘Q’ are given by (xp , yp , zp ) and (xq, yq, zq), the distance vector (or separation vector) is the displacement from ‘P’ to ‘Q’ as shown. rpq = rq - rp = (xq - xp )a x + (yq - yp )a y +(zq - zp )a z. Note that point P is not a vector, only position vector of ‘P’ i.e., rp is only the vector. 2. Vector Multiplication: The product of two vectors A and B is either a scalar or vector depending on the manner how they are multiplied. There are two types of vector multiplication.  Scalar or Dot Product: A. B.  Vector or Cross Product: A X B. Dot Product: The dot product of two vectors A and B written as A. B is defined geometrically as the product of magnitudes of A and B and the cosine of the smaller angle between them. A.B = |A| |B| cos θAB The dot product of two vectors yields a scalar. If A = (Ax,, Ay, Az) and B = (Bx,By, Bz) , then A.B = Ax Bx + Ay By + Az Bz which is obtained by multiplying A and B component by component. It is to be noted that scalar or dot product obeys the commutative law. i.e. A.B = B.A Two vectors A and B are said to be orthogonal i.e. perpendicular if their dot product is zero. i.e. A.B = 0 Conversely, we can say that if the dot product of two vectors A and B is zero, and neither of the vectors is zero, then they are perpendicular. The scalar product of a vector with itself yields the square of its magnitude.

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Thus, A.A = | A| | A| cos 0° = A2 The scalar product of a vector and a unit vector yields the component of the vector in the unit vector direction. Thus, we have Ax .A = | Ax| | A| cos α = Ax Ay .A = | Ay| | A| cos β = Ay Az .A = | Az| | A| cos γ = Az Where cos α, cos β and cos γ are the direction cosines with α the angle between A and xaxis, β the angle between A and y-axis and γ the angle between A and z-axis. The dot product obeys the Distributive Law. A.(B+C) = A.B + A.C The dot product of unit vectors in the rectangular coordinate system is as follows: ax.ay = ay.az = a z.ax = 0 but, ax.ax = ay. ay = az. az = 1 In other words, the scalar or dot product of one coordinate unit vector with a different one is always zero, while the scalar product of a unit vector with itself is always unity ( for orthogonal system). The Line Integral: An important application of the scalar product involves the line integral. Suppose we move along a curved path from point P 1 to P2 in a radial force field F acting on an object in the r (radial) direction. At any point ‘P’ the product of a path length dL and the component of F parallel to it is given by F cos θ dL = FL dL where, FL = component of F in the direction of path and θ = angle between positive directions of path and F. From the figure, it is clear that the component of dL in the r (and F) direction is given by dr = cos θ dL Using vector notation, we have F.dL = Fcos θ dL= FL dL = F dr. 7

where dL = vector incremental length (magnitude dL in the direction of path). The product of a force F and a distance dr represents an incremental amount of work dW done by the force F in moving an object a distance cos θ dL = dr. Thus, dW = F.dL = F cos θ dL If the path is broken up into segments parallel and perpendicular to F, we note from above equation that contributions to the work occur only for the segments parallel to F (θ = 0°) with no work for the segments perpendicular to F (θ = 90°). Summing up the contributions of the segments parallel to F, we obtain the total work W between the end points of the point. For finite length segments dL, this value is approximate. As dL → 0, it becomes exact as given by P2 (end) ∫ F .dL P1 (start) where, P1 = starting point (lower limit of integration) P2 = end point (upper limit of integration) The formulation of above equation is called a line integral giving the total work W done by F on an object (equals energy imparted to the object) moved over the path from P 1 to P2 . W =

Ex.1: Work in Linear Field A force field F is in the x-direction and increases linearly with distance x. Thus, F = x a x. Find the work done by the force F in moving an object from a point x = 1 to a point x = 2. Sol.

F.dL = x dx, so 2 2 W = ∫F.dL = ∫x dx = ½ x2 | = 3/2 1 1

If F is in Newton and x is in meters, the work is in joules. Ex.2: Work in Radial Field A radial force field F decreases with distance as given by F = r-2 a r. Find the work done in moving from a point at r = √2 to a point at r = 2√2 by a direct path and one following rectangular coordinates. Sol.

1) F.dL = r-2 a r 2√2 W = ∫ F.dL = ∫ r-2 dr √2

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2√2 = - (1/r)| √2 = 1/(2√2) 2) Instead of integrating along the direct path from r = √2 (at (x, y) = (1, 1)) to r = 2√2 (at (x, y) = (2, 2)), let us follow a rectangular coordinate path from x, y = 1, 1 to x, y = 2, 1 (constant y) and then from x, y = 2, 1 to x, y = 2, 2 (constant x). Thus for constant y path, we have (F.dL)y = a r/(x2 + y2 ) . a x dx where a r . a x = cos 45° = x/(x2 + y2 ) and (F.dL)x = x/((x2 + y2 ) √(x2 + y2 )) Since x = y and dx = dy, the x-constant and y-constant terms are equal. Thus, the total work is twice the work for y-constant path. 2 i.e. W = ∫F.dL = 2∫ x/((x2 + y2 ) √(x2 + y2 )) dx 1 2 2 2 2 (3/2) 2 =2∫ x/(2x ) dx = (2/√8) ∫ (1/x )dx = (-1/√2) (1/x) | 1 1 1 = 1/(2√2) Note: For a vector like F, the line integral depends only on the end points so we could follow any path. Further, if we integrate F around a closed path, starting say at x, y = 1, 1 and ending back at x, y = 1, 1, the result is zero. Thus, ∮ F. dL = 0 where ∮ indicates integration around a closed path. Any field for which the line integral around a closed path is zero is called a conservative or lamellar field. Not all fields are lamellar. Proble ms: 1. Find the work required to move a 5kg mass from x = 0, y = 0 to x = 8, y = 7m against a force F = x2 a x N. 2. Find the work done in lifting a 6 – tonne (m = 6 x 103 kg) satellite to GSO, height = 37,000 km. The force of gravity is F = (G ms me )/r2 a r N 9

Where, ms = mass of satellite, kg. me = mass of earth = 6 x 1024 kg. r = distance of satellite from centre of earth, m G = gravitational constant = 6.67 x 10-11 Nm2 kg-2 The circumference of earth = 40, 000 km. Cross Product: The cross product of two vectors A and B, written as A X B, is a vector quantity whose magnitude is the area of the parallelepiped formed by A and B and is in the direction of advance of a right – handed screw as A is turned into B. Thus, A X B = AB sinθAB a n Where, an is a unit vector normal to the plane containing A and B. θAB is the smaller angle between the two vectors. The direction of a n is taken as the direction of the right thumb when the fingers of the right hand rotate from A to B. If A = (Ax, ,Ay, Az) and B = (Bx,By, Bz) , then AXB= A X B = (Ay Bz - Az By ) a x + (Az Bx - Ax Bz) a y + (Ax By - Ay Bx ) a z which is obtained by crossing the terms in cyclic permutation. Reversing the order of the vectors A and B results in a unit vector in the opposite direction, and we see that the cross product is not commutative, for B X A = - (A X B). If the definition of cross product is applied to the unit vectors a x and a y, we find ax X ay = az for each vector has unit magnitude, the two vectors are perpendicular, and the rotation of ax into ay indicates the positive z – direction by the definition of a right – handed coordinate system. In a similar way, ay X az = ax az X ax = ay Cross Product has the following basic properties: 1. It is not commutative: AXB≠BXA It is anti – commutative: A X B = - (B X A) 10

2. It is not associative: A X (B X C) ≠ (A X B) X C 3. It is distributive: AX (B + C) = A X B + A X C 4. A X A = 0 A simple example of the use of the cross product may be taken from geometry or trigonometry. To find the area of the parallelogram, the product of the lengths of the two adjacent sides is multiplied by the sine of the angle between them. Using vector notation for the two sides, we then may express the (scalar) area as the magnitude of A X B or |A X B|. Ex.1. Vector A = 8 a x + 3 a y - 10 a z and vector B = -15 a x + 6 a y + 17 a z. Find A X B Sol.

A X B = (Ay Bz - Az By ) a x + (Az Bx - Ax Bz) a y + (Ax By - Ay Bx ) a z ⇒ A X B = (3*17 + 10*6) a x + (10*15 – 8*17) a y + (8*6 + 3*15) a z = 111 a x + 14 a y + 93 a z

Proble ms: 1) Vector A = 20 a y – 5 a z and B = -6 a x + 14 a y. Find A X B. 2) The three vertices of a triangle are located at A(6, -1, 2), B(-2, 3, -4) and C(-3, 1, 5). Find a) RAB X RAC b) the area of the triangle. c) A unit vector perpendicular to the plane in which the triangle is located. Scalar Triple Product: Just as multiplication of two vectors gives a scalar or vector result, multiplicat ion of three vectors A, B and C gives a scalar or vector result depending on how the vectors are multiplied. Thus, we have scalar or vector triple product. Given three vectors A, B and C, we define the scalar triple product as A.(B X C) = B. (C X A) = C. (A X B) obtained in cyclic permutation. If A = (Ax, ,Ay, Az), B = (Bx,, By, Bz) and C = (Cx,,Cy, Cz), then A.(B X C) is the volume of a parallelepiped having A, B and C,

i.e. A.(B X C) =

Since the result of this vector multiplication is scalar, it is called scalar triple product. Vector Triple Product: For vector A, B and C, we define the vector triple product as, A X (B X C) = B. (A.C) - C. (A.B) 11

It should be noted that (A . B) C ≠ A (B.C) but (A . B) C = C (A.B)

Practice Exercise: 1. Find the vector A directed from (2, -4, 1) to (0, -2, 0) in Cartesian coordinates and find the unit vector along A. 2. Given the vector field, F = 0.4(y – 2x) a x – (200/(x2 +y2 +z2 )) a z. a) Evaluate |F| at P(-4, 3, 5) b) Find a unit vector specifying the direction of F at P. Describe the locus of all points for which c) Fx = 1 d) |Fz| = 2 3. Let E = 3 a y + 4 a z and F = 4 a x - 10 a y + 5 a z. a) Find the component of E along F b) Determine a unit vector perpendicular to both E and F. 4. Show that a = (4, 0, -1), b = (1, 3, 4) and c = (-5, -3, -3) form the sides of a triangle. Is this a right – angled triangle? Calculate the area of the triangle. 5. Given points A(2, 5, -1), B(3, -2, 4) and C(-2, 3, 1), find a) RAB, R AC. b) The angle between RAB and RAC. c) The length of projection of RAB on RAC. d) The vector projection of RAB on RAC 6. A triangle is defined by the three points A (2, -5, 1), B (-3, 2, 4) and C (0, 3, 1). Find a) RBC X RBA b) The area of the triangle c) A unit vector perpendicular to the plane in which the triangle is located. 7. Show that A = 4 a x - 2 a y - a z and B = a x + 4 a y - 4a z are perpendicular. 8. Given A = a x + a y, B = a x + 2a z, C = 2a y + a z, find (A X B) X C and compare it with A X (B X C). 9. Using the vectors given above find A.BXC and compare it with A X B.C

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