46-55in13.1

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 CHAPTER 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 ±gure. 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 =h 7 x 0 , 8 y 0 i AB 5 2 , 4 2 i=h 3 , 2 i The vectors and 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 We denote the points in the ±gure 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: =h− 1 ( 3 ), b 2 2 , b 2 i DC 2 a , 3 1 2 a , 2 i Since = , the two vectors have the same components. That is, 2 = 2 a b 2 = 2 a = 0 b = 4 48. Let v = 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 d ) v w (i) CB (ii) CA (iii) BC (iv) BA (a) w has the same length as w and points in the opposite direction. Hence: w = .
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SECTION 13.1 Vectors in the Plane (ET Section 12.1) 305 C A w (b) v has the same length as v and points in the opposite direction. Hence:
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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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