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Unformatted text preview: 1 t, z 1 2t 73. Point: Q 0, 0, 0 Plane: 2x 3y z 12 0 Normal to plane: n 10, contradiction
\ 71. Writing the equation of the line in parametric form and substituting into the equation of the plane we have: x 21 1 3t 3t, y 3 1 1 2t 2t, z 10, 3 1 t 2, 3, 1 Point in plane: P 6, 0, 0 Vector PQ
\ Therefore, the line does not intersect the plane. 6, 0 0 12 14 6 14 7 D PQ n n 75. Point: Q 2, 8, 4 Plane: 2x y z 5 2, 1, 1 77. The normal vectors to the planes are n1 1, 3, 4 and n2 1, 3, 4 . Since n1 n2, the planes are parallel. Choose a point in each plane. P Q
\ Normal to plane: n
\ Point in plane: P 0, 0, 5 Vector: PQ
\ 10, 0, 0 is a point in x 6, 0, 0 is a point in x
\ 3y 3y 4z 4z 4 26 10. 6. 2 26 13 2, 8, 11 6 1 11 6 6 PQ 4, 0, 0 , D D PQ n n PQ n1 n1 79. The normal vectors to the planes are n1 3, 6, 7 and n2 6, 12, 14 . Since n2 2n1, the planes are parallel. Choose a point in each plane. P Q
\ 4, 0, 1 is the direction vector for the line. 81. u Q 1, 5, 2 is the given point, and P 2, 3, 1 is on the line. Hence, PQ 3, 2, 3 and
\ \ 0, 1, 1 is a point in 3x 6y 12y 7z 14z 1. PQ 25.
\ 25 , 0, 0 is a point in 6x 6 25 , 1, 6
\ u i 3 4 j 2 0 149 17 k 3 1 2, 2533 17 9, 8 PQ D 1 27 2 94 27 2 94 27 94 188 a, b, c, D PQ u u PQ n1 n1 83. The parametric equations of a line L parallel to v and passing through the point P x1, y1, z1 are x x1 at, y y1 bt, z z1 ct. 85. Solve the two linear equations representing the planes to find two points of intersection. Then find the line determined by the two points. The symmetric equations are x a x1 y b y1 z c z1 . Section 10.6 87. (a) Sphere x x2 89. (a) z 3 y
2 2 Surfaces in Space 249 (b) Parallel planes y z
2 2 6x 2 z 4y 1.09y 1985 5 2 16 22 0 4x 3y z 10 ± 4 n 10 ± 4 26 10z 28.7 Year 1.83x 1980 16.16 1990 9.81 1994 8.60 1995 8.42 1996 8.27 1997 8.23 z (approx.) 14.23 (b) An increase in x or y will cause a decrease in z. In fact, any increase in two variables will cause a decrease in the third. (c)
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z (0, 0, 28.7) (15.7, 0, 0)
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x (0, 26.3, 0)
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y 91. True Section 10.6
1. Ellipsoid Matches graph (c) 7. z 3 Surfaces in Space
3. Hyperboloid of one sheet Matches graph (f)
z 5. Ellipt...
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This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL

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