642 CHAPTER 9 Polar Coordinates; Vectors Figure 60 a2 a2 b1 b2 b2 a1 y v 1 w (a2, b2) (a1, b1) (a1 1 a2, b1 1 b2) O x (a) Illustration of property (2) aa1 b1 ab1 a1 y av (a1, b1) (aa1, ab1) O x (b) Illustration of property (4), a . 0 b1 b1 y P1 5 (a1, b1) O x (c) a1 a2 1 1 b2 1 Illustration of property (5): || v || 5 Distance from O to P1 || v || 5 w v v v Adding and Subtracting Vectors If = + = 〈 〉 v i j 2 3 2, 3 and = − = 〈 − 〉 w i j 3 4 3, 4,find: (a) +v w (b) −v w Solution EXAMPLE 3 (a) ( ) ( ) ( ) ( ) +=++−=++− =− v w i j i j i j i j 2 3 3 4 2 3 3 4 5 or ( ) + =〈 〉+〈 −〉=〈 + + − 〉=〈 −〉 v w 2,3 3, 4 2 3,3 4 5, 1 (b) ( ) ( ) ( ) ( ) [ ] −= +−− =−+−− =−+ v w i j i j i j i j 2 3 3 4 2 3 3 4 7 or ( ) − =〈 〉−〈 −〉=〈 − − − 〉=〈− 〉 v w 2,3 3, 4 2 3,3 4 1, 7 4 Find a Scalar Multiple and the Magnitude of a Vector Finding Scalar Multiples and Magnitudes of Vectors If = + = 〈 〉 v i j 2 3 2, 3 and = − = 〈 − 〉 w i j 3 4 3, 4 , find: (a) v3 (b) −v w 2 3 (c) v Solution EXAMPLE 4 (a) ( ) = + = + v i j i j 3 3 2 3 6 9 or = 〈 〉 = 〈 〉 v3 32,3 6,9 (b) ( ) ( ) −= +− −=+−+ =−+ v w i j i j i j i j i j 2 3 2 2 3 3 3 4 4 6 9 12 5 18 or ( ) − =〈 〉−〈−〉=〈 〉−〈−〉 =〈− −− 〉=〈− 〉 v w 2 3 22,3 33, 4 4,6 9, 12 4 9, 6 12 5, 18 (c) = + = + = v i j 2 3 2 3 13 2 2 Now Work PROBLEMS 35 AND 43 5 Find a Unit Vector Recall that a unit vector u is a vector for which = u 1. In many applications, it is useful to be able to find a unit vector u that has the same direction as a given vector v.
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