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homework01_w12 - 50 N exerted at an angle of 20 ◦ above...

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Math 215 Homework Set 1: §§ 12.1–12.3 Winter 2012 Most of the following problems are modified versions of the problems from your text book, Multivariable Calculus , 7th ed., by James Stewart. Your solution to each problem should be complete, show all work, and be written in complete sentences where appropriate. For Maple problems, include a print-out that shows all of the work and graphs that you generated in Maple to solve the problem, in addition to any work you may have done by hand. 12.1.1: Find the equation of the sphere that contains the points (4 , 5 , 3) , (9 , 0 , 3) and (0 , 0 , 0) , if its center lies in the xz -plane. 12.1.2: Sketch and describe in words the region in R 3 represented by the inequality x 2 + y 2 + z 2 < 4 z. 12.1.3: Write equations or inequalities that describe the solid lower half of a ball of radius 2 centered at the origin. 12.2.1: Consider the vectors a = h- 3 , 2 , 1 i and b = h 0 , - 2 , 3 i . Sketch each of the following quantities: (a) a - 2 b (b) - 2 a + 3 b 12.2.2: A child pulls a sled through the snow on a level path with a force of
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Unformatted text preview: 50 N exerted at an angle of 20 ◦ above the horizontal. Find the horizontal and vertical components of the force. 12.2.3: (a) Consider the line which is tangent to the curve y = 2 cos 2 x at the point ( π/ 6 , 1) . Find all unit vectors that are parallel to this line. (b) Find all unit vectors that are perpendicular to this line. (c) Sketch the curve y = 2 cos 2 x , the tangent line described above, and all of the unit vectors described above. Draw all of the unit vectors starting at the point ( π/ 6 , 1) . 12.3.1: (a) Under what circumstances is comp a b = comp b a ? (b) Under what circumstances is proj a b = proj b a ? 12.3.2: Use a scalar projection to show that the distance from a point ( x 1 ,y 1 ) to the line ax + by + c = 0 is | ax 1 + by 1 + c | √ a 2 + b 2 ....
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