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Copyright 2002, 1963 by The McGrawHill Companies, Inc. Click Here for Terms of Use. in Visual Studio .NET
Copyright 2002, 1963 by The McGrawHill Companies, Inc. Click Here for Terms of Use. QR Code Reader In .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. QR Code 2d Barcode Maker In .NET Using Barcode encoder for .NET Control to generate, create QR Code JIS X 0510 image in .NET applications. CHAP. 7] Read QR In VS .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in VS .NET applications. Generate Barcode In VS .NET Using Barcode creator for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. VECTORS
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Fig. 74 The di erence of vectors A and B, represented by A B, is that vector C which added to B gives A. Equivalently, A B may be de ned as A B . If A B, then A B is de ned as the null or zero vector and is represented by the symbol 0. This has a magnitude of zero but its direction is not de ned. The expression of vector equations and related concepts is facilitated by the use of real numbers and functions. In this context, these are called scalars. This special designation arises from application where the scalars represent object that do not have direction, such as mass, length, and temperature. Multiplication of a vector A by a scalar m produces a vector mA with magnitude jmj times the magnitude of A and direction the same as or opposite to that of A according as m is positive or negative. If m 0, mA 0, the null vector. ALGEBRAIC PROPERTIES OF VECTORS The following algebraic properties are consequences of the geometric de nition of a vector. Problems 7.1 and 7.2.) (See VECTORS
[CHAP. 7
If A, B and C are vectors, and m and n are scalars, then 1. 2. 3. 4. 5. A B B A A B C A B C m nA mn A n mA m n A mA nA m A B mA mB Commutative Law for Addition Associative Law for Addition Associative Law for Multiplication Distributive Law Distributive Law Note that in these laws only multiplication of a vector by one or more scalars is de ned. On Pages 153 and 154 we de ne products of vectors. LINEAR INDEPENDENCE AND LINEAR DEPENDENCE OF A SET OF VECTORS A set of vectors, A1 ; A2 ; . . . ; Ap , is linearly independent means that a1 A1 a2 A2 ap Ap ap Ap 0 if and only if a1 a2 ap 0 (i.e., the algebraic sum is zero if and only if all the coe cients are zero). The set of vectors is linearly dependent when it is not linearly independent. UNIT VECTORS Unit vectors are vectors having unit length. If A is any vector with length A > 0, then A=A is a unit vector, denoted by a, having the same direction as A. Then A Aa. RECTANGULAR (ORTHOGONAL) UNIT VECTORS The rectangular unit vectors i, j, and k are unit vectors having the direction of the positive x, y, and z axes of a rectangular coordinate system [see Fig. 75]. We use righthanded rectangular coordinate systems unless otherwise speci ed. Such systems derive their name from the fact that a rightthreaded screw rotated through 908 from Ox to Oy will advance in the positive z direction. In general, three vectors A, B, and C which have coincident initial points and are not coplanar are said to form a righthanded system or dextral system if a rightthreaded screw rotated through an angle less than 1808 from A to B will advance in the direction C [see Fig. 76 below]. Fig. 75 Fig. 76 CHAP. 7]

