Fundamental Electromagnetics [ Chapter 2: Vector Algebra ] Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-6919-2160 1
Key Points Basic concept of scalars and vectors What is unit vector? Vector addition and subtraction Position and distance vectors Vector multiplication Dot product Cross product Components of a vector Dept. Information and Communication Eng. 2
Scalars and Vectors Scalars Definition Magnitude only Vectors Magnitude & Direction Scalar quantities: Time, Mass, Distance, Temperature, etc Vector quantities: Velocity, Force, Acceleration, etc Field: A function that specifies a particular quantity everywhere in a region Dept. Information and Communication Eng. 3
Scalars and Vectors Scalar field: Is a function that specifies a scalar quantity everywhere in a region Height of a mountain Sound intensity in a theater Is just one where a quantity in space is represented by numbers, such as this temperature map Can express as a function T( x, y) Dept. Information and Communication Eng. 4
Scalars and Vectors Vector field: Is a function that specifies a vector quantity everywhere in a region Gravitational force on a body in a space Wind map in the atmosphere Dept. Information and Communication Eng. 5
Vector Notation Vectors show magnitude and displacement, drawn as a ray : A Dept. Information and Communication Eng. 6
Unit vector Vector A : Magnitude of A : A or A Direction of A : unit vector whose magnitude is unity. That is, Or can use Bold-Faced Type like A aa, A A a A A In Cartesian(or rectangular) coordinates, it represents as Ax, Ay, Az or Axa x Aya y Aza z where Ax, Ay,and Az are components of A in x, y and z directions, respectively. And, a, a,and a are unit vectors in those directions. x y z a A a A A (Graphical Representation) a A A A Aa A Dept. Information and Communication Eng. 7
Unit vector The magnitude and unit vector of vector A are then Axax Ayay Aa z z 2 2 2 A Ax Ay A aa z 2 2 2 A A A x y z a x a z a y A a x x A A a z z A a y y Dept. Information and Communication Eng. 8
Unit vector Example 2.1: Describe the vectors P and R, shown in the following figures z z P y 3a x R 4a y y x P 1a y x R 3a 4a x y Dept. Information and Communication Eng. 9
Unit vector Example 2.2: Describe the vectors R and R 1 2, shown in the following figures R 2 R, 1 aax bay R aa ba ca 2 x y z R 1 Dept. Information and Communication Eng. 10
Vector addition and subtraction Two vectors A and B add together to obtain a new vector C : C A B if A A, A, A and B B, B, B, then x y z x y z C A B a A B a A B a x x x y y y z z z Vector subtraction is similarly carried out as D AB A( B) A B a A B a A B a x x x y y y z z z Dept. Information and Communication Eng. 11
Vector addition and subtraction Vectors may be added graphically, head to tail. Dept. Information and Communication Eng. 12
Vector addition and subtraction Special case of vector addition Add the negative of the subtracted vector Continue with standard vector addition procedure A B A B Dept. Information and Communication Eng. 13
Vector addition and subtraction Example 2.3: x y z (a) the component of A along a, y (b) the magnitude of 3A B, (c) a unit vector along A 2B If A 10a 4a 6a and B a a, find Solution: 2 x y (a) Ay 4 (b) Since 3AB 3(10, 4,6) (2,1,0) (28, 13,18), we have 2 2 2 3AB 28 ( 13) 18 1277 (c) Letting C A2 B(10, 4, 6) (4, 2, 0) (14, 2, 6), a unit vector along C is C (14, 2, 6) ac 2 2 2 C 14 ( 2) 6 Dept. Information and Communication Eng. 14
Vector addition and subtraction Example 2.4: x z (a) A B, (b) 5A B, (c ) The component of A along a y, (d) A unit vector parallel to 3A B Given vector A a 3a and B 5a 2a 6a, determine Solution: (a) (b) (c) (d) A unit vector parallel to this vector is x y z Dept. Information and Communication Eng. 15
Vector addition and subtraction MatLab Lesson: MatLab scripts to calculate a vector magnitude function y=magvector(r) % Calculates the magnitude of a Cartesian vector R y=sqrt(r(1)^2+r(2)^2+r(3)^2); MatLab scripts to calculate the unit vector function y=unitvector(r) % Calculates the unit vector of a Cartesian vector R y=r/magvector(r); Find the unit vector of vector A 10a 4a 6a using MatLab >> A=[10-4 6]; >> unitvector(a) x y z Dept. Information and Communication Eng. 16
Visual EMT using MatLab Show the graphical representation of addition and subtraction of two vectors in Example 2.3 Run vectoralg.m! Dept. Information and Communication Eng. 17
Position and distance vectors A point P may be described by (x, y, z) Then, the position vector r p of point P is the directed distance from the origin to P, that is, r OP xa ya za p x y z As example, a position vector 3a 4a 5a is shown as x y z a a z x a y r xa ya za xyz,, p x y z r p Dept. Information and Communication Eng. 18
Position and distance vectors Distance vector: displacement from one point to another if two points P and Q are given as xp, yp, zp and, the distance vector is xq, yq, zq r r r x x a y y a z z a PQ Q P Q P x Q P y Q P z Q r Q r r r PQ Q P r PQ r P r P P r Q r a 4a 2a PQ x y z Dept. Information and Communication Eng. 19
Position and distance vectors Example 2.5: Given points P 1, 3,5, Q 2, 4,6 and R 0,3,8, find: (a) the position vectors of P and R, (b) the distance vector r QR, (c) the distance between P and Q Solution: (a) (b) (c) Dept. Information and Communication Eng. 20
Relative velocity in 1-D Example 2.6: Woman walks with a velocity of 1.0 m/s along the aisle of a train that is moving with a velocity of 3.0 m/s. What is the woman s velocity? Solution: For passenger sitting in a train: 1.0 m/s For bicyclist standing: Dept. Information and Communication Eng. 21
Relative velocity in 1-D Cyclist: frame of reference A Moving train: frame of reference B In 1-D motion, position of P relative to frame of reference A is given by distance x P/A Position of P relative to frame of reference B is given by distance x P/B Distance from origin A to origin B is given by x B/A Thus, xp/ A xb/ A xp/ B Velocity v P/A of P relative to frame A is the derivative of x P/A with respect to time dx dt dxp dt dxb dt P / A / B / A v v v P/ A P/ B B/ A 1.0 m/ s3.0 m/ s 4.0 m/ s Dept. Information and Communication Eng. 22
Relative velocity in 2-D Example 2.7: A river flows SE at 10 km/hr, and a boat flows upon it. A man walks upon the deck at 2 km/hr to the perpendicular direction. Find the velocity of the man with respect to the earth. Solution: Since the velocity of boat is 10 cos 45 sin 45 u a a 7.071( a a ) km/hr b x y x y u m u ab u b Dept. Information and Communication Eng. 23
Relative velocity in 2-D, and the velocity of the man with respect to the boat (relative velocity) is 2 u cos45 a sin45 a 1.414( a a ) km/hr m x y the absolute velocity of the man is u u u 5.657a 8.485a km/hr ab m b x y 10.2 j56.3 e that is, 10.2 km/hr at 56.3 o south of east x y Dept. Information and Communication Eng. 24
Relative velocity in 2-D Example 2.8: The velocity of the boat relative to the water is 4.0 m/s, directed perpendicular to the current The river is 1.8 km wide and the velocity of the water relative to the shore is 2.0 m/s How far upstream is the boat when it reaches the opposite shore? Solution: Dept. Information and Communication Eng. 25
Vector multiplication There are two types of vector multiplication: Scalar (or dot) product: A B Vector (or cross) product: A B Dot product: A B= A B cosq AB Here AB is the smaller angle between two vectors, and the result is scalar. If A A,, x Ay Az and B B x, By, Bz, then A B= Ax Bx + AyBy + AzBz using axay = ayaz = azax = 0 (Orthogonal) a a = a a = a a = 1 x x y y z z % Matlab script >> dot(a,b) Dept. Information and Communication Eng. 26
Vector multiplication a B AB Example 2.9: (Graphical Representation) Work of a force acting on a body when the body is moved by a small distance x Work done by the force is F D x= Fcosq Dx ( ) (Projection of the force on the x-axis) ( Product between the components of the same direction ) A B a (Projection= B cos AB ) Dept. Information and Communication Eng. 27
Vector multiplication Example 2.10: Find the work done against gravity to move a 10 kg baby from the point (2,3) to the point (5,7)? Solution: We have that the force vector is 10 F mg 9.8a y And the displacement vector is x 52 a 73 a 3a 4a The work is the dot product x y x y F D x = -98a 3a + 4a =-392 J ( y) ( x y) Notice the negative sign verifies that the work is done against gravity Hence, it takes work of 392 J to move the baby y (2,3) (5,7) x Dept. Information and Communication Eng. 28
Vector multiplication Example 2.11: If A a 3a and B 5a 2a 6a, find x Solution: z Using the dot product, x y z AB Thus, % Matlab script A = [1 0 3]; B = [5 2-6]; Num = dot(a,b); Den=sqrt(sum(A.^2))*sqrt(sum(B.^2)); Theta_AB = (180/pi)*acos(Num/Den) Dept. Information and Communication Eng. 29
Vector multiplication There are instances in real world where the product of two vectors is another vector Torque: The torque (turning force) vector lies in a direction perpendicular to the plane formed by the position vector ( r ) and the force vector ( F ) The torque is the vector (or cross) product of the position vector and the force vector Dept. Information and Communication Eng. 30
Vector multiplication Cross Product: % Matlab script where a AB ABsin AB an >> cross(a,b) n is a unit vector normal to the area of parallelogram, and the result is vector a n (Right hand rule) A B B (Right handed screw rule) A B AB ABsin AB = area of parallelogram a n B A A Dept. Information and Communication Eng. 31
Vector multiplication If A A x, Ay, Az and B B, then x, By, Bz ax ay az AB A A A x y z Bx By Bz A B A B a A B A B a A B A B a y z z y x z x x z y x y y x z Basic properties: a x a a z y a a z y a x a a a a a a a a a a a a a a a x y z y x y z x z y z x y x z (More easily remembered form) AB B A A BC AB C A B C AB AC Dept. Information and Communication Eng. 32,,, (Commutative Law) (Associative Law) (Distributive Law)
Vector multiplication Example 2.12: Let A 3ay 4az and B 4a, x 10ay 5az (a) Find the vector component of A along B A B (b) Determine a unit vector perpendicular to both and A ab = Acos( qab ) = AB B (a) Since A ( ), using, we have B = Aa B B = A ab a a B B B AB B 104, 10,5 A 0.2837a 0.7092a 0.3546a 2 B 141 Solution: B x y z (b) Using ax ay az AB 0 3 4 55,16, 12, 4 10 5 0.9398,0.2734, 0.205 aa B Dept. Information and Communication Eng. 33
Vector multiplication Example 2.13: Show that vector a 4,0, 1, b 1,3, 4, and form the sides of a triangle. Is this a right angle? Calculate the area of the triangle Solution: Since, it is a right angle triangle Area = c 5, 3, 3 % Matlab script a = [4 0-1]; b = [1 3 4]; c = cross(a,b); Area = 1/2*sqrt(sum(c.^2)); Rectangle Area Dept. Information and Communication Eng. 34
Visual EMT using MatLab Plot the trajectory of a particle moving in Cartesian coordinate in terms of vector notations Run rectcoord.m! Dept. Information and Communication Eng. 35
Visual EMT using MatLab Let s A a and 2 x (a) Plot the Dot-product (b) Plot the Cross-product 2 2 x y B sin xy ax e a AB A B y Run vecalgebra.m! Dept. Information and Communication Eng. 36
Vector multiplication Scalar triple product: Given three vectors A, B, and C, the scalar triple product is A B C = B C A = C A B ( ) ( ) ( ) if A A, A, A, B B, B, B and C C, C, C, then x y z x y z A A A A B C = B B B ( ) x y z x y z C C C x y z ( Volume of the parallelepiped ) x y z Dept. Information and Communication Eng. 37
Vector multiplication a n Height = A A a n C Area = B C Vector triple product: Given three vectors A, B, and C, the vector triple product is A B C = B AC -C AB B ( ) ( ) ( ) Dept. Information and Communication Eng. 38
Components of a vector A direct application of vector product is to determine the projection (or component) of a vector in a desired direction Given a vector A, the scalar component along vector B is AB = AcosqAB = A ab cosqab = A ab The vector component of A along vector B is A Aa Aa a A B B B B B B Dept. Information and Communication Eng. 39 A B A A B AB
Components of a vector Division of vectors does not consider because it is undefined Coordinate components of a vector: A y : Component along y-direction a y a y a x a x A x : Component along x-direction Dept. Information and Communication Eng. 40
Components of a vector There are two methods of vector addition Graphical : represent vectors as scaled-directed line segments; attach tail to head Analytical : resolve vectors into x and y components; add components R AB, R, x Ax Bx Ry Ay By Ax Acos A, Ay Asin A B B cos, B B sin x B y B Dept. Information and Communication Eng. 41
Application of vectors Example 2.14: Derive the cosine formula Solution: 2 2 2 a b c 2bccosA From the triangle of figure, we know that ab c 0, that is, b c a Thus, it becomes ( ) ( ) 2 a = aa= b+ c b+ c = bb+ cc+ 2bc = + - 2 2 b c 2bccosA c a b a b Dept. Information and Communication Eng. 42
Application of vectors Example 2.15: Derive the law of sines for a triangle using vectors Solution: B C A We have Because, we can write B B 0 or BC BA Thus, BC A sin Similarly, we have sin BAsin C sin A B C sin sin sin B C A 0 C A B 180 Dept. Information and Communication Eng. 43
Homework Assignments Problem 2.1: Find the unit vector along the line joining point (2, 4, 4) to point (-3, 2, 2). Problem 2.2: Let A 2ax 5ay 3 az, B 3ax 4ay, and (a) Determine A 2B (b) Calculate A 5C (c) For what values of k is kb 2? (d) Find A B / A B Problem 2.3: Show that ( ) ( ) 2 2 ( A B) + A B = ( AB) 2 C a a a x y z Dept. Information and Communication Eng. 44
Homework Assignments Problem 2.4: P1, 2,3, P 5, 2,0 P Points 1 2, and 3 2, 7, 3 form a triangle in space. Find the three angles of the triangle. Problem 2.5: E and F are vector fields given by E 2xax ay yzaz and 2 F xya y a xyza. Determine x y z (a) E at (1, 2, 3) (b) The component of E along F at (1, 2, 3) (c) A vector perpendicular to both E and F at (0, 1, -3) whose magnitude is unity. Dept. Information and Communication Eng. 45
Homework Assignments Problem 2.6: Find the vector R using the component of vectors Dept. Information and Communication Eng. 46