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The sum or resultant of vectors A and B of Fig. 7-3(a) below is a vector C formed by placing the initial point of B on the terminal point of A and joining the initial point of A to the terminal point of B [see Fig. 7-3(b) below]. The sum C is written C A B. The de nition here is equivalent to the parallelogram law for vector addition as indicated in Fig.7-3(c) below. Extensions to sums of more than two vectors are immediate. For example, Fig. 7-4 below shows how to obtain the sum or resultant E of the vectors A, B, C, and D.
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B+C+
Fig. 7-4
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. 7-5]. We use right-handed rectangular coordinate systems unless otherwise speci ed. Such systems derive their name from the fact that a right-threaded 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 right-threaded screw rotated through an angle less than 1808 from A to B will advance in the direction C [see Fig. 7-6 below].
Fig. 7-5
Fig. 7-6
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