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6. jA Bj the area of a parallelogram with sides A and B. 7. If A B 0 and neither A nor B is a null vector, then A and B are parallel.
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TRIPLE PRODUCTS Dot and cross multiplication of three vectors, A, B, and C may produce meaningful products of the form A B C; A B C ; and A B C . The following laws are valid: 1. A B C 6 A B C in general 2. A B C B C A C A B volume of a parallelepiped having A, B, and C as edges, or the negative of this volume according as A, B, and C do or do not form a righthanded system. If A A1 i A2 j A3 k, B B1 i B2 j B3 k and C C1 i C2 j C3 k, then    A1 A2 A3    6 A B C  B1 B2 B3     C1 C2 C3  3. A B C 6 A B C 4. A B C A C B A B C A B C A C B B C A (Associative Law for Cross Products Fails)
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The product A B C is called the scalar triple product or box product and may be denoted by ABC . The product A B C is called the vector triple product. In A B C parentheses are sometimes omitted and we write A B C. However, parentheses must be used in A B C (see Problem 7.29). Note that A B C A B C. This is often expressed by stating that in a scalar triple product the dot and the cross can be interchanged without a ecting the result (see Problem 7.26).
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AXIOMATIC APPROACH TO VECTOR ANALYSIS From the above remarks it is seen that a vector r xi yj zk is determined when its 3 components x; y; z relative to some coordinate system are known. In adopting an axiomatic approach, it is thus quite natural for us to make the following De nition. A three-dimensional vector is an ordered triplet of real numbers with the following properties. If A A1 ; A2 ; A3 and B B1 ; B2 ; B3 then 1. A B if and only if A1 B1 ; A2 B2 ; A3 B3 2. A B A1 B1 ; A2 B2 ; A3 B3 3. A B A1 B1 ; A2 B2 ; A3 B3 4. 0 0; 0; 0 5. mA m A1 ; A2 ; A3 mA1 ; mA2 ; mA3 In addition, two forms of multiplication are established. 6. A B A1 B1 A2 B2 A3 B3 p q 7. Length or magnitude of A jAj A A A2 A2 A2 1 2 3 8. A B A2 B3 A3 B2 ; A3 B1 A1 B3 ; A1 B2 A2 B1 Unit vectors are de ned to be 1; 0; 0 ; 0; 1; 0 ; 0; 0; 1 and then designated by i; j; k, respectively, thereby identifying the components axiomatically introduced with the geometric orthonormal basis elements. If one wishes, this axiomatic formulation (which provides a component representation for vectors) can be used to reestablish the fundamental laws previously introduced geometrically; however, the
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primary reason for introducing this approach was to formalize a component representation of the vectors. It is that concept that will be used in the remainder of this chapter. Note 1: One of the advantages of component representation of vectors is the easy extension of the ideas to all dimensions. In an n-dimensional space, the component representation is A A1 ; A2 ; . . . ; An An exception is the cross-product which is speci cally restricted to three-dimensional space. There are generalizations of the cross-product to higher dimensional spaces, but there is no direct extension.) Note 2: The geometric interpretation of a vector endows it with an absolute meaning at any point of space. The component representation (as an ordered triple of numbers) in Euclidean three space is not unique, rather, it is attached to the coordinate system employed. This follows because the components are geometrically interpreted as the projections of the arrow representation on the coordinate directions. Therefore, the projections on the axes of a second coordinate system rotated (for example) from the rst one will be di erent. (See Fig. 7-10.) Therefore, for theories where groups of coordinate systems play a role, a more adequate component de nition of a vector is as a collection of ordered triples of numbers, each one identi ed with a coordinate system of the group, and any two related by a coordinate transformation. This viewpoint is indispensable in Newtonian mechanics, electromagnetic theory, special relativity, and so on.
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Fig. 7-10
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VECTOR FUNCTIONS If corresponding to each value of a scalar u we associate a vector A, then A is called a function of u denoted by A u . In three dimensions we can write A u A1 u i A2 u j A3 u k. The function concept is easily extended. Thus, if to each point x; y; z there corresponds a vector A, then A is a function of x; y; z , indicated by A x; y; z A1 x; y; z i A2 x; y; z j A3 x; y; z k. We sometimes say that a vector function A de nes a vector eld since it associates a vector with each point of a region. Similarly,  x; y; z de nes a scalar eld since it associates a scalar with each point of a region.
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LIMITS, CONTINUITY, AND DERIVATIVES OF VECTOR FUNCTIONS Limits, continuity, and derivatives of vector functions follow rules similar to those for scalar functions already considered. The following statements show the analogy which exists. 1. The vector function represented by A u is said to be continuous at u0 if given any positive number , we can nd some positive number  such that jA u A u0 j <  whenever ju u0 j < . This is equivalent to the statement lim A u A u0 . The derivative of A u is de ned as
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