Explanation Given: The parametric equations of the first line are: x − 2 − 3 = y − 2 6 = z − 3 The parametric equations of the second line are: x − 3 2 = y + 5 = z + 2 4 Formula used: The angle between two lines is given by cos θ = v 1 ⋅ v 2 ‖ v 1 ‖ ‖ v 2 ‖ Calculation: First, convert the symmetric equations of both the lines into parametric equations by equating the equations with t and s respectively. Then, Equations of the first line will be, x − 2 − 3 = y − 2 6 = z − 3 = t On simplifying, x = 2 − 3 t , y = 2 + 6 t , z = t + 3 And, the equation of the second line will be x − 3 2 = y + 5 = z + 2 4 = s On simplifying, x = 2 s + 3 , y = s − 5 , z = 4 s − 2 Since, the lines intersect to each other at a point Then the intersect point will be common for both the lines. Then, the coordinates x , y and z for both the lines will be equal at the point of intersection. Now, equate x , y and z coordinates of both the lines. Then, For x coordinate, 2 − 3 t = 2 s + 3 …… (1) For y coordinate, 2 + 6 t = s − 5 …… (2) For z coordinate, t + 3 = 4 s − 2 …… (3) Since, by z coordinate, the relation between t and s can be easily determined. Then, Form the equation (3), t + 3 = 4 s − 2 t = 4 s − 5 Now, the value of s can be determined by putting t = 4 s − 5 in equation (1) then, 2 − 3 ( 4 s − 5 ) = 2 s + 3 2 − 12 s + 15 = 2 s + 3 − 14 s = − 14 s = 1 Therefore, the value of s is 1 . Now, with the value of s , then find the value of t , Since, t = 4 s − 5 , Then substitute the value of s and then find the value of t . Then, t = 4 s − 5 = 4 ( 1 ) − 5 = 4 − 5 = − 1 Now, substitute t = − 1 and s = 1 in the equation of first line and second line respectively. Then, The x coordinate of for the first line is, x = 2 − 3 t = 2 + 3 = 5 The y coordinate of the first line is, y = 2 + 6 t = 2 − 6 = − 4 The z coordinate of the first line is, z = t + 3 = − 1 + 3 = 2 So, from the above calculation the coordinates for the first line are ( 5 , − 4 , 2 )

7th Edition

ISBN: 9781337678445

Author: Larson, Edward

Publisher: CENGAGE L

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Chapter 11.5, Problem 36E

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To calculate: The point of intersection of lines x−2−3=y−26=z−3, x−32=y+5=z+24 if they intersect and the angle between the lines.

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Chapter 11.5, Problem 35E

Chapter 11.5, Problem 37E

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Chapter 11 Solutions

CALCULUS EARLY TRANSCENDENTAL FUNCTIONS

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The point of intersection of lines x − 2 − 3 = y − 2 6 = z − 3 , x − 3 2 = y + 5 = z + 2 4 if they intersect and the angle between the lines.

Solution Summary: The author calculates the angle between two lines by equating the symmetric equations of both the lines.

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To calculate: The point of intersection of lines x−2−3=y−26=z−3, x−32=y+5=z+24 if they intersect and the angle between the lines.

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Chapter 11 Solutions