0 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.01 Fall 2005 Review: Vector Analysis A.1 Vectors 1 A.1.1 Introduction 1 A.1.2 Properties of a Vector 1 A.1.3 Application of Vectors: 6 A.2 Dot Product 10 A.2.1 Introduction 10 A.2.2 Definition 10 A.2.3 Properties of Dot Product 11 A.2.4 Vector Decomposition and the Dot Product: 12 A.3 Cross Product 13 A.3.1 Introduction 14 A.3.2 Definition: Cross Product 14 A.3.3 Right hand Rule for the Direction of Cross Product 14 A.3.4 Properties of the Cross Product: 16 A.3.5 Vector Decomposition and the Cross Product 17 Example: Torque 181 Vector Analysis A.1 Vectors A.1.1 Introduction Some physical quantities like the mass or the temperature at some point only have magnitude. We can represent these quantities by number alone (with the appropriate units) and so we call them scalars. There are other physical quantities that have magnitude and direction. Their magnitude can stretch or shrink, and their 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 either one. Position, displacement, velocity, acceleration, force, momentum, and torque are all physical quantities that can be mathematically represented by vectors. One of the most difficult problems in understanding physics is learning how to represent these physical quantities as mathematical vectors. We shall begin by defining precisely what we mean by a vector. A.1.2 Properties of a Vector A vector can be thought of as an object that has direction and magnitude. We denote a vector by the symbol !Athe magnitude of !A by the symbol A where A = Ax2+ Ay2+ Az2(see Eq 1.1.16). 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) Scalar Multiplication of Vectors: vectors can be multiplied by real numbers.2 Let !A be a vector. Let c be a real positive number. Then the multiplication of !A by c is a new vector that we denote by the symbol c!A. The magnitude of c!A is c times the magnitude of !A (Figure A.1.2a), cA = Ac (A.1.1) Since c > 0, the direction of c!A is the same as the direction of !A. However, if c < 0, then the direction of c!A points in the opposite direction of !A (Figure A.1.2b). Figure A.1.2a and A.1.2a Multiplication of vectors by numbers (2) Vector Addition: Vectors can be added. Let !A and !B be two vectors. We define a new vector, !C =!A +!B, the `vector addition’ of !A and !B, by a geometric construction. Draw the arrow that represents !A. Place the tail of the arrow that represents !B at the tip of the arrow for !A as shown in Figure A.1.3a. The arrow that starts at the tail of !A and goes to the tip of !B is defined to be the `vector addition’ !C =!A +!B. There is an equivalent construction for the law of vector addition. The vectors !A and !B 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 !C =!A +!B (Figure A.1.3b).3 Figure A.1.3a and A.1.3b Geometric sum of vectors Vector addition satisfies the following four properties: (3) Commutivity: The order of adding vectors does not matter !A +!B =!B +!A (A.1.2) Our geometric definition for vector addition satisfies property (3) since in the parallelogram representation for the addition of vectors, it doesn’t matter which side you start with as seen in Figure A.1.4. Figure A.1.4 Commutative property of vector addition (4) Associativity: When adding three vectors, it doesn’t matter which two you start with (!A +!B) +!C =!A + (!B +!C) (A.1.3) In Figure A.1.5a, we add (!A +!B) +!C, while in Figure A.1.5b we add !A + (!B +!C). We arrive at the same new vector in either case. Figure A.1.5a and A.1.5b Associative law4 (5) Identity Element for Vector Addition: There is a unique vector, !0, that acts as an identity element for vector addition. This means that for all vectors !A, !A +!0 =!0 +!A =!A (A.1.4) (6) Inverse element for Vector Addition: For every vector !A, there is a unique inverse vector !1( )!A " !!A (A.1.5) such that !A + !!A( )=!0 This means that the vector !!A has the same magnitude as !A, | A!"|=| !A!"|= A, but they point in opposite directions (Figure A.1.6). Figure A.1.6 additive inverse Scalar multiplication of vectors satisfies the following four properties: (7) Associative Law for Scalar Multiplication: The order of multiplying numbers is doesn’t matter. Let b and c be real numbers. Then b(c!A) = (bc)!A (A.1.6) (8) Distributive Law for Vector Addition: Vector addition satisfies a distributive law for multiplication by a number. Let c be a real number. Then5 c(!A +!B) = c!A + c!B (A.1.7) Figure A.1.7 illustrates this property. Figure A.1.7 Distributive Law for vector addition (9) Distributive Law for Scalar Addition: The multiplication operation also satisfies a distributive law for the addition of numbers. Let b and c be real numbers. Then (b + c)!A = b!A + c!A (A.1.8) Our geometric definition of vector addition satisfies this condition as seen in Figure A.1.8. Figure A.1.8 Distributive law for scalar multiplication (10) Identity Element for Scalar Multiplication: The number 1 acts like an identity element for multiplication, 1!A =!A (A.1.9)6 A.1.3 Application of Vectors: When we apply vectors to physical quantities it’s nice to keep in the back of our minds all these formal properties. However from the physicist’s point of view, we are interested in representing physical quantities like displacement, velocity, acceleration, force, impulse, momentum, torque, and angular momentum as vectors. We can’t add force to velocity or
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