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# homework 1 - 0.0 1 Hand-in Assignment 1 Solutions(Math...

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0.0. 1 Hand-in Assignment 1 Solutions (Math 20300 DD, PP, ST) Your Name: Foster, Antony Your Student ID: 0000 (only last 4 digits) Your Instructor: Prof. A. Foster Due Date: Friday, February 8, 2008. SECTION 13.1 (pages 829–834): DO Exercises 1, 3, 4, 5, 12, 18, 19, 36, 40, and 42. Formulas Used: Distance Formula in Three Dimensions : The distance k P 1 P 2 k between the points P 1 ( x 1 , y 1 , z 1 ) and P 2 ( x 2 , y 2 , z 2 ) in 3-Dimensional real space or R 3 is k P 1 P 2 k = p ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 . (1) Equation of a Sphere : An equation of a sphere with center C ( α, β, γ ) and radius ρ is ( x - α ) 2 + ( y - β ) 2 + ( z - γ ) 2 = ρ 2 . (2) In particular, if the center is the origin O (0 , 0 , 0), then an equation of the sphere is x 2 + y 2 + z 2 = ρ 2 (3) Volume of a spherical cap : The volume V of a cap of a sphere centered at the origin O (0 , 0 , 0) with radius ρ and height h < ρ of the cap (measured from the endpoint of a diameter of the sphere) is V = 1 3 πh 2 (3 ρ - h ) (can be derived using calculus) . (4) Distance from a Point to a Plane : Let P be a plane with equation ax + by + cz + d = 0 and P ( x 0 , y 0 , z 0 ) is a point not in the plane P then the distance d ( p, P ) from the point P to the plane P is given by d ( P, P ) = | ax 0 + by 0 + cz 0 + d | a 2 + b 2 + c 2 . (5)

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2 0. Exercise 1 : Suppose you start at the origin, move along the x -axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position? Solution. Without any drawings we start at the origin O (0 , 0 , 0) and move along the x -axis a distance of 4 units in the positive direction. This puts us at the point A (4 , 0 , 0). From A we move downward a distance of 3 units, this puts our final position at the point B (4 , 0 , - 3). Note this is a point in the xz -plane. tu Exercise 3 : Which of the points P (6 , 2 , 3) , Q ( - 5 , - 1 , 4), and R (0 , 3 , 8) is closest to the xz -plane? Which point lies in the yz -plane? Solution. Of the three points P, Q and R we can see that Q is closest to the xz -plane. The distance to the xz -plane from a point A ( x, y, z ) in space is determined by | y | (i.e., absolute value of the y -coordinate of the point).
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homework 1 - 0.0 1 Hand-in Assignment 1 Solutions(Math...

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