Explanation Description: Generally, the two line which meet in space will intersect each other and making an angle in degrees at the point of intersection. Otherwise, Figure 1 shows the intersection of two lines in space. In Figure 1, the two lines AB and CD are intersect each other in space with some angle in θ ° . For example: Intersection of two planes: Consider the two planes are 3 x − 7 y − z = 15 and 2 x + 3 y − 2 z = 5 . Find the parametric equations of the line. Since the line of intersection of two planes is perpendicular to both planes’ normal vectors n 1 and n 2 , and therefore, n 1 × n 2 = | i j k 3 − 7 − 1 2 3 − 2 | = ( 14 + 3 ) i − ( − 6 + 2 ) j + ( 9 + 14 ) k = 17 i + 4 j + 23 k The general expression for the parametric equations for a line through P 0 ( x 0 , y 0 , z 0 ) parallel to a nonzero vector v = v 1 i + v 2 j + v 3 k is, x = x 0 + t v 1 , y = y 0 + t v 2 , z = z 0 + t v 3 (1) To find the point on the line, just take any point common to the two planes and choose z = 0 arbitrary in the planes equation and solve for x and y . Therefore, the two planes equation becomes, 3 x − 7 y = 15 (2) 2 x + 3 y = 5 (3) On solving equation (1) and (2), x = 80 23 y = − 15 23 Hence, the point on the line is ( 80 23 , − 15 23 , 0 )

Thomas' Calculus: Early Transcendentals (14th Edition)

14th Edition

ISBN: 9780134439020

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 12, Problem 15GYR

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Explain the intersection of two lines in space. Also find the intersection of two planes, a line and a plane with some examples.

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Chapter 12, Problem 14GYR

Chapter 12, Problem 16GYR

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Thomas' Calculus: Early Transcendentals (14th Edition)

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