math103Exam1Sec2Solutions

math103Exam1Sec2Solutions - Math 103 section 2 - Exam 1...

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Math 103 section 2 - Exam 1 Instructions: You are not permitted to use a calculator or any other materials on this exam. Show all your work on the test. You may use the back of the pages if needed. Simplify all answers. Give exact answers (i.e. fractions, radicals, etc.) unless instructed otherwise. Sign the Honor Pledge after you ﬁnish the exam. I have neither given nor received any unauthorized help on this test, and I have conducted myself within the guidelines of the university honor code. Name and section: Pledge: Question: 1 2 3 4 5 6 7 8 Total Points: 21 5 12 10 6 10 14 6 84 Score:

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1. (21 points) Given the points P (1 , - 1 , 1), Q (2 , 0 , 3), R (1 , 1 , 2), S (2 , 1 , 4). (a) Compute the angle between ~ PQ and ~ PS . Solution: ~ PQ = h 1 , 1 , 2 i and ~ PS = h 1 , 2 , 3 i So, θ = cos - 1 (1+2+6) 1+1+4 1+4+9 = cos - 1 9 2 21 (b) Give the equation of the plane containing the points P , Q , and S . Solution: n = ~ PQ × ~ PS = ± ± ± ± ± ± i j k 1 1 2 1 2 3 ± ± ± ± ± ± = h- 1 , - 1 , 1 i So, the equation of the plane is - ( x - 1) - ( y + 1) + ( z - 1) = 0 or simpliﬁed: - x - y + z = 1 (c) Find the symmetric equations of the line perpendicular to the plane thru P,Q,S which passes through R . Solution: Direction of the line will be n as computed above. - ( x - 1) = - ( y - 1) = ( z - 2) (d) Compute the volume of the parallelepiped spanned by ~ PQ , ~ PR , and ~ PS . Solution:
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This note was uploaded on 04/19/2010 for the course MATH 103 taught by Professor Hain during the Fall '08 term at Duke.

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math103Exam1Sec2Solutions - Math 103 section 2 - Exam 1...

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