SECTION 9.4 Vectors 641 Figure 58 y v 5 85, 49 (5, 4) P2 5 (4, 6) P1 5 (21, 2) O 5 5 x Finding a Position Vector Find the position vector of the vector = PP v 1 2 if ( ) = − P 1, 2 1 and ( ) = P 4, 6 . 2 Solution EXAMPLE 2 By equation (1), the position vector equal to v is ( ) = 〈 − − − 〉 = 〈 〉 v 4 1 , 6 2 5, 4 See Figure 58. Figure 59 Unit vectors i and j y j i (0, 1) (1, 0) x In Words To add two vectors, add corresponding components. To subtract two vectors, subtract corresponding components. Two position vectors v and w are equal if and only if the terminal point of v is the same as the terminal point of w . This leads to the following theorem: THEOREM Equality of Vectors Two vectors v and w are equal if and only if their corresponding components are equal. That is, = 〈 〉 = 〈 〉 a b a b v w If , and , 1 1 2 2 = = = a a b b v w then if and only if and . 1 2 1 2 We now present an alternative representation of a vector in the plane that is common in the physical sciences. Let i denote the unit vector whose direction is along the positive x -axis; let j denote the unit vector whose direction is along the positive y -axis. Then = 〈 〉 i 1, 0 and = 〈 〉 j 0, 1 , as shown in Figure 59. Any vector = 〈 〉 a b v , can be written using the unit vectors i and j as follows: =〈 〉=〈 〉+〈 〉= + a b a b a b v i j , 1, 0 0, 1 The quantities a and b are called the horizontal and vertical components of v , respectively. For example, if = 〈 〉 = + v i j 5, 4 5 4 , then 5 is the horizontal component and 4 is the vertical component. Now Work PROBLEM 29 3 Add and Subtract Vectors Algebraically The sum, difference, scalar multiple, and magnitude of algebraic vectors are defined in terms of their components. DEFINITION Suppose = + = 〈 〉 a b a b v i j , 1 1 1 1 and = + = 〈 〉 a b a b w i j , 2 2 2 2 are two vectors, and α is a scalar. Then ( ) ( ) +=+ ++ =〈+ +〉 a a b b a a b b v w i j , 1 2 1 2 1 2 1 2 (2) ( ) ( ) −=− +− =〈− −〉 a a b b a a b b v w i j , 1 2 1 2 1 2 1 2 (3) α α α α α ( ) ( ) = + = 〈 〉 a b a b v i j , 1 1 1 1 (4) = + a b v 1 2 1 2 (5) These definitions are compatible with the geometric definitions given earlier in this section. See Figure 60 on the next page.
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