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Unformatted text preview: Oliver Knill, Harvard Summer school, Summer 2009 Homework for Chapter 1. Geometry and Space Section 1.1: Space, distance, geometrical objects 1) (Geometrical objects) Describe and sketch the set of points P = ( x, y, z ) in R 3 repre sented by a) 9 y 2 + 4 z 2 = 81 b) x/ 7 − y/ 11 − z/ 13 = 1 c) ( x + 3) 2 = 25 d) d ( P, (1 , , 0)) + d ( P, (0 , 1 , 0) = 10. e) xy = 4 f) x 2 + ( y − 2) 2 = 0 g) x 2 − y 2 = 0 h) x 2 = y . Solution: a) An elliptical cylinder with the x axis as a center width 3 and height 4.5. b) A plane through the points (7 , , 0) , (0 , − 11 , 0) , (0 , , − 13). c) The union of two planes x = − 8 and y = − 2. d) An ellipsoid with focal points (1 , , 0) and (0 , 1 , 0). e) A hyperbolic cylinder. f) A line x = 0 , y = 2, an axes parallel to the z axes. g) A union of two planes which are perpendicular to each other and intersect in the zaxis. The z − trace forms 45 degree angles with the x and y axes. h) A cylindrical paraboloid. 2) (Distances) a) Find the distance from the point P = (3 , 2 , 5) i) to the yaxis. ii) to the xzcoordinate plane. Solution: (i) A general point ( x, y, z ) has distance √ x 2 + z 2 from the yaxis and distance y from the xzcoordinate system. In our case, the point P has distance √ 9 + 25 from the yaxis. (ii) The distance is 2 from the xzcoordinate plane. 3) (Distances) Below you see two rectangles, one of area 8 · 8 = 64, an other of area 65 = 13 · 5. But there are triangles or trapezoids which match. What is going on? Look at distances. 1 LParen1 0,0 RParen1 LParen1 13,0 RParen1 LParen1 13,5 RParen1 LParen1 0,5 RParen1 LParen1 0,8 RParen1 LParen1 8,3 RParen1 LParen1 5,2 RParen1 LParen1 5,5 RParen1 LParen1 0,0 RParen1 LParen1 8,0 RParen1 LParen1 8,8 RParen1 LParen1 0,8 RParen1 LParen1 8,3 RParen1 LParen1 5,3 RParen1 LParen1 0,3 RParen1 LParen1 3,8 RParen1 Solution: Not all points which appear to be on lines are really on lines. Look at the second picture. The line from (0 , 0) to (5 . 2) to (8 . 3) and (13 , 5) is broken. The distances do not add up. For example, √ 29 + √ 10 = 8 . 547 ... does not add up to √ 73 = 8 . 544 ... . If you draw everything right, there will be a narrow piece of area which is left out. This explains the larger area in the second picture. 4) (Spheres, traces) Find an equation of the sphere with center ( − 1 , 8 , − 5) and radius 6. Describe the traces of this surface, its intersection with each of the coordinate planes. Solution: ( x + 1) 2 + ( y − 8) 2 + ( z + 5) 2 = 36 is the equation of the sphere. The traces are obtained by putting x = 0 or y = 0 or z = 0. x = 0 gives the yztrace: ( y − 8) 2 + ( z + 5) 2 = 35 is a circle. y = 0 gives the xztrace: ( x + 1) 2 + ( z + 5) 2 = 36 − 64 is the empty set....
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 Spring '10
 Knill
 Calculus, Geometry

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