A.1 VectorsIntroductionProperties of a VectorApplication of VectorsIn terms of magnitudes and angles, we haveDot ProductIntroductionDefinitionProperties of Dot ProductVector Decomposition and the Dot ProductCross ProductDefinition: Cross ProductFigure A.3.1 Cross product geometry.Right-hand Rule for the Direction of Cross ProductFigure A.3.2 Right-Hand Rule.Properties of the Cross ProductVector Decomposition and the Cross ProductMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2004 Review A: Vector Analysis A.1 Vectors………………………………………………………………………….. A-2 A.1.1 Introduction………………………………………………………………. A-2 A.1.2 Properties of a Vector…………………………………………………….. A-2 A.1.3 Application of Vectors…………………………………………………….A-6 A.2 Dot Product………………………………………………………………... A-10 A.2.1 Introduction ...…………………………………………………………....A-10 A.2.2 Definition………………………………………………………………... A-11 A.2.3 Properties of Dot Product……………………………………………….. A-12 A.2.4 Vector Decomposition and the Dot Product ……………………………. A-12 A.3 Cross Product …………………………………………………………….. A-14 A.3.1 Definition: Cross Product ………………………………………………. A-14 A.3.2 Right-hand Rule for the Direction of Cross Product …………………… A-15 A.3.3 Properties of the Cross Product ………………………………………… A-16 A.3.4 Vector Decomposition and the Cross Product …………………………. A-17 A-1Vector Analysis A.1 Vectors A.1.1 Introduction Certain physical quantities such as mass or the absolute temperature at some point only have magnitude. These quantities can be represented by numbers alone, with the appropriate units, and they are called scalars. There are, however, other physical quantities which have both magnitude and direction; the magnitude can stretch or shrink, and the direction can reverse. These quantities can be added in such a way that takes into account both direction and magnitude. Force is an example of a quantity that acts in a certain direction with some magnitude that we measure in newtons. When two forces act on an object, the sum of the forces depends on both the direction and magnitude of the two forces. Position, displacement, velocity, acceleration, force, momentum and torque are all physical quantities that can be represented mathematically by vectors. We shall begin by defining precisely what we mean by a vector. A.1.2 Properties of a Vector A vector is a quantity that has both direction and magnitude. Let a vector be denoted by the symbol. The magnitude of AGAG is | A≡A|G. We can represent vectors as geometric objects using arrows. The length of the arrow corresponds to the magnitude of the vector. The arrow points in the direction of the vector (Figure A.1.1). Figure A.1.1 Vectors as arrows. There are two defining operations for vectors: (1) Vector Addition: Vectors can be added. Let and be two vectors. We define a new vector, AGBG=+CABGGG, the “vector addition” of and , by a geometric construction. Draw the arrow that represents . Place the AGBGAG A-2tail of the arrow that represents BG at the tip of the arrow for AG as shown in Figure A.1.2(a). The arrow that starts at the tail of AG and goes to the tip of is defined to be the “vector addition”. There is an equivalent construction for the law of vector addition. The vectors and can be drawn with their tails at the same point. The two vectors form the sides of a parallelogram. The diagonal of the parallelogram corresponds to the vector , as shown in Figure A.1.2(b). BG=+CABGGGAGBG=+CABGGG Figure A.1.2 Geometric sum of vectors. Vector addition satisfies the following four properties: (i) Commutivity: The order of adding vectors does not matter. +=+ABBAGGGG (A.1.1) Our geometric definition for vector addition satisfies the commutivity property (i) since in the parallelogram representation for the addition of vectors, it doesn’t matter which side you start with as seen in Figure A.1.3. Figure A.1.3 Commutative property of vector addition (ii) Associativity: When adding three vectors, it doesn’t matter which two you start with () ()++=+ +AB CA BCGGG GGG (A.1.2) In Figure A.1.4(a), we add ( )++AB CGGG, while in Figure A.1.4(b) we add . We arrive at the same vector sum in either case. ( )++ABCGGG A-3Figure A.1.4 Associative law. (iii) Identity Element for Vector Addition: There is a unique vector, , that acts as an identity element for vector addition. 0G This means that for all vectors , AG +=+ =A00A AGGGG G (A.1.3) (iv) Inverse element for Vector Addition: For every vector AG, there is a unique inverse vector ()1−≡−AAGG (A.1.4) such that ()+−=AA0GGG This means that the vector − has the same magnitude asAGAG, ||| |A=−=AAJGJG, but they point in opposite directions (Figure A.1.5). Figure A.1.5 additive inverse. (2) Scalar Multiplication of Vectors: Vectors can be multiplied by real numbers. A-4Let be a vector. Let c be a real positive number. Then the multiplication of by c is a new vector which we denote by the symbol cAGAGAG. The magnitude of c is c times the magnitude of (Figure A.1.6a), AGAG cA Ac= (A.1.5) Since , the direction of is the same as the direction ofc > 0 cAGAG. However, the direction of is opposite of (Figure A.1.6b). c− AGAG Figure A.1.6 Multiplication of vector AG by (a) , and (b) . 0c > 0c−< Scalar multiplication of vectors satisfies the following properties: (i) Associative Law for Scalar Multiplication: The order of multiplying numbers is doesn’t matter. Let b and c be real numbers. Then ()() ( ) ()bc bc cb cb===AAAAGGG Gc (A.1.6) (ii) Distributive Law for Vector Addition: Vector addition satisfies a distributive law for multiplication by a number. Let c be a real number. Then ()cc+=+AB A BGGGG (A.1.7) Figure A.1.7 illustrates this property. A-5Figure A.1.7 Distributive Law for vector addition. (iii) Distributive Law for Scalar Addition: The multiplication operation also satisfies a
View Full Document
Unlocking...