Explanation 1) Concept: i. To find the direction vector of lines L 1 & L 2 and identify the lines are skew lines, then find the common normal vector to finding the distance between lines L 1 & L 2 ii. The distance between the parallel planes a x + b y + c z + d 1 = 0 & a x + b y + c z + d 2 = 0 is, D = d 1 - d 2 a 2 + b 2 + c 2 2) Given: Let L 1 be the line through 1 , 2 , 6 and the point 2 , 4 , 8 Let L 2 be the line of intersection of the planes P 1 and P 2 P 1 x - y + 2 z + 1 = 0 P 2 is the plane through the points 3 , 2 , - 1 , 0 , 0 , 1 , 1 , 2 , 1 3) Calculation: Let L 1 be the line through 1 , 2 , 6 and the point 2 , 4 , 8 Then, the direction vector of the line L 1 is 2 , 4 , 8 - 1 , 2 , 6 = 1 , 2 , 2 The equation of plane P 1 is, x - y + 2 z + 1 = 0 The normal vector of plane P 1 is n 1 = 1 , - 1 , 2 Now, P 2 is the plane through the points 3 , 2 , - 1 , 0 , 0 , 1 , 1 , 2 , 1 So, the vector form of points is, Consider a = 3 , 2 , - 1 - 0 , 0 , 1 = 3 , 2 , - 2 b = 1 , 2 , 1 - 0 , 0 , 1 = 1 , 2 , 0 So, the normal vector of P 2 is, n 2 = 3 , 2 , - 2 × 1 , 2 , 0 = i j k 3 2 - 2 1 2 0 = 4 i - 2 j + 4 k That is, n 2 = 4 , - 2 , 4 It can be written as

Bundle: Calculus, 8th + Enhanced WebAssign - Start Smart Guide for Students + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Multi-Term

8th Edition

ISBN: 9781305525924

Author: James Stewart

Publisher: Cengage Learning

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Chapter 12.5, Problem 80E

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The distance between the lines L1& L2

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Chapter 12.5, Problem 79E

Chapter 12.5, Problem 81E

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Bundle: Calculus, 8th + Enhanced WebAssign - Start Smart Guide for Students + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Multi-Term

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