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46-55in13.1 - 304 C H A P T E R 13 V E C T O R G E O M E T...

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304 C H A P T E R 13 VECTOR GEOMETRY (ET CHAPTER 12) 46. What are the coordinates of the point P in the parallelogram in Figure 25(A)? SOLUTION We denote by A , B , C the points in the figure. x y C (7, 8) P ( x 0 , y 0 ) B (5, 4) A (2, 2) Let P = ( x 0 , y 0 ) . We compute the following vectors: −→ PC = 7 x 0 , 8 y 0 −→ AB = 5 2 , 4 2 = 3 , 2 The vectors −→ PC and −→ AB are equivalent, hence they have the same components. That is: 7 x 0 = 3 8 y 0 = 2 x 0 = 4 , y 0 = 6 P = ( 4 , 6 ) 47. What are the coordinates a and b in the parallelogram in Figure 25(B)? x y x y (2, 2) (A) P (5, 4) (7, 8) (2, 3) ( 3, 2) ( a , 1) ( 1, b ) (B) FIGURE 25 SOLUTION We denote the points in the figure by A , B , C and D . x y C (2, 3) A ( 3, 2) D ( a , 1) B ( 1, b ) We compute the following vectors: −→ AB = − 1 ( 3 ), b 2 = 2 , b 2 −→ DC = 2 a , 3 1 = 2 a , 2 Since −→ AB = −→ PC , the two vectors have the same components. That is, 2 = 2 a b 2 = 2 a = 0 b = 4 48. Let v = −→ AB and w = −→ AC , where A , B , C are three distinct points in the plane. Match (a)–(d) with (i)–(iv). ( Hint: Draw a picture.) (a) w (b) v (c) w v (d) v w (i) −→ C B (ii) −→ C A (iii) −→ BC (iv) −→ B A SOLUTION (a) w has the same length as w and points in the opposite direction. Hence:
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