Explanation Given: The two lines r ( t ) = 〈 4 + 5 t , − 2 t , 1 + 3 t 〉 and R ( s ) = 〈 10 s , 6 + 4 s , 4 + 6 s 〉 . Definition used: The two lines are said to be parallel lines if their direction vectors are proportional to each other. The two lines are said to be intersecting lines if they have common point. The two lines are said to be skew lines if the lines are neither parallel nor intersecting lines. Calculation: Consider the line r ( t ) = 〈 4 + 5 t , − 2 t , 1 + 3 t 〉 , simplify and compare with the line equation r ( t ) = r 0 + t u as follows, r ( t ) = 〈 4 + 5 t , − 2 t , 1 + 3 t 〉 = 〈 4 , 0 , 1 〉 + 〈 5 t , − 2 t , 3 t 〉 = 〈 4 , 0 , 1 〉 + t 〈 5 , − 2 , 3 〉 Thus, the vector u is 〈 5 , − 2 , 3 〉 . Consider the line R ( s ) = 〈 10 s , 6 + 4 s , 4 + 6 s 〉 , simplify and compare with the line equation R ( s ) = R 0 +s v as follows, R ( s ) = 〈 10 s , 6 + 4 s , 4 + 6 s 〉 = 〈 0 , 6 , 4 〉 + 〈 10 s , 4 s , 6 s 〉 = 〈 0 , 6 , 4 〉 + s 〈 10 , 4 , 6 〉 Thus, the vector v is 〈 10 , 4 , 6 〉 . Observe that the vector v is not proportional to the vector u . v = 〈 10 , 4 , 6 〉 = 2 〈 5 , 2 , 3 〉 ≠ 2 u Therefore, the given lines are not parallel. Set the lines r ( t ) = R ( s ) . 〈 4 + 5 t , − 2 t , 1 + 3 t 〉 = 〈 10 s , 6 + 4 s , 4 + 6 s 〉 (1) Equate the x -coordinates on both sides and simplify

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 11.5, Problem 52E

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The nature of the lines.

Solution Summary: The author explains that the lines are skew lines. They are parallel lines if their direction vectors are proportional to each other.

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