13.5.Ex47-63

13.5.Ex47-63 - 388 C H A P T E R 13 V E C T O R G E O M E T...

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388 CHAPTER 13 VECTOR GEOMETRY (ET CHAPTER 12) SOLUTION The trace of the plane x + y = 4onthe xz -plane is the set of all points that satisfy both the equation of the given plane and the equation y = 0ofthe -plane. That is, the set of all points ( x , 0 , z ) such that x + 0 = 4, or x =− 4. This is a vertical line in the -plane. 41. x + y = 4, yz The trace of the plane x + y = -plane is the set of all points that satisfy both the equation of the plane and the equation x = -plane. That is, the set of all points ( 0 , y , z ) such that 0 + y = 4, or y = 4. This is a vertical line in the -plane. 42. Does the plane x = 5 have a trace in the -plane? Explain. The -plane has the equation x = 0, hence the x -coordinates of the points in this plane are zero, whereas the x -coordinates of the points in the plane x = 5 are 5. Thus, the two planes have no common points. 43. Give equations for two distinct planes whose trace in the xy -plane has equation 4 x + 3 y = 8. The -plane has the equation z = 0, hence the trace of a plane ax + by + cz = 0inthe -plane is obtained by substituting z = 0 in the equation of the plane. Therefore, the following two planes have trace 4 x + 3 y = 8 in the -plane: 4 x + 3 y + z = 8 ; 4 x + 3 y 5 z = 8 44. Find parametric equations for the line through P 0 = ( 3 , 1 , 1 ) perpendicular to the plane 3 x + 5 y 7 z = 29. We need to ±nd a direction vector for the line. Since the line is perpendicular to the plane 3 x + 5 y 7 z = 29, it is parallel to the vector n =h 3 , 5 , 7 i normal to the plane. Hence, n is a direction vector for the line. The vector parametrization of the line is, thus, r ( t ) 3 , 1 , 1 i+ t h 3 , 5 , 7 i This yields the parametric equations x = 3 + 3 t , y 1 + 5 t , z = 1 7 t 45. Find all planes in R 3 whose intersection with the -plane is the line r ( t ) = t h 2 , 1 , 0 i . The intersection of the plane + + = d with the -plane is obtained by substituting z = equation of the plane. This gives the line + = d ,inthe -plane. We ±nd the equation of the line l ( t ) = t h 2 , 1 , 0 i . On this line we have x = 2 t y = t y = 1 2 x x 2 y = 0 We thus must have d = 0and b a 2, a ±= 0. That is, d = 0, b 2 a , a ±= 0. Notice that c can have any value. Hence, the planes are 2 ay + = 0 , a ±= 0 46. Find all planes in R 3 whose intersection with the -plane is the line with equation 3 x + 2 z = 5. The intersection of the plane + + = d with the -plane is obtained by substituting y = equation of the plane. This gives the following line in the -plane: + = d This is the equation of the line 3 x + 2 z = 5 if and only if for some λ ±= 0, a = 3 , c = 2 , d = 5 Notice that b can have any value. The planes are thus ( 3 ) x + + ( 2 ) z = 5 , ±= 0 . 47. Let P be the plane n · h x , y , z i = d ,where n ±= 0 ,andlet P 1 be the parallel plane n · h x , y , z i = d 1 (Figure 10). (a) Show that the line through n intersects P at the terminal point of the vector µ d k n k e n . (b) Show that the distance between P and P 1 is | d d 1 | k n k .
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SECTION 13.5 Planes in Three-Space (ET Section 12.5) 389 y x z n n x , y , z = d 1 n x , y , z = d d d 1 ⏐⏐ n P P 1 FIGURE 10 SOLUTION We place the vector n so that the ray through n passes through the origin.
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This homework help was uploaded on 04/22/2008 for the course MATH 32A taught by Professor Gangliu during the Winter '08 term at UCLA.

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13.5.Ex47-63 - 388 C H A P T E R 13 V E C T O R G E O M E T...

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