Explanation Given information: The equations r 1 ( t ) = 〈 5 t , 2 t − 1 , 2 t − 2 〉 and r 2 ( t ) = 〈 t − 5 , − t + 4 , t − 7 〉 . Concept used: The parametric equations of the line passing though the point P = ( x 0 , y 0 , z 0 ) and parallel to the vector v = 〈 a , b , c 〉 are, x = x 0 + a t y = y 0 + b t z = z 0 + c t } ,.... ( 1 ) Two lines intersect at a point where their respective coordinates are equal to each other.. The general equation of a plane is, A ( x − x 0 ) + B ( y − y 0 ) + C ( z − z 0 ) = 0 ,.... ( 2 ) where A , B and C are the components of the normal vector on the plane and ( x 0 , y 0 , z 0 ) is the point lying on the plane. Consider the line r 1 ( t ) = 〈 5 t , 2 t − 1 , 2 t − 2 〉 . Then, x = 5 t 1 y = 2 t 1 − 1 z = 2 t 1 − 2 } ,.... ( 3 ) Here v 1 = 〈 5 , 2 , 2 〉 . Use similar argument for r 2 ( t ) = 〈 t − 5 , − t + 4 , t − 7 〉 . x = t 2 − 5 y = − t 2 + 4 z = t 2 − 7 } ,.... ( 4 ) Here v 2 = 〈 1 , − 1 , 1 〉 . Equate equations ( 3 ) and ( 4 ) . 5 t 1 = t 2 − 5 ,.... ( 5 ) 2 t 1 − 1 = − t 2 + 4 ,.... ( 6 ) 2 t 1 − 2 = t 2 − 7 ,.... ( 7 ) Consider equation ( 5 ) . t 1 = t 2 − 5 5 Substitute t 2 − 5 5 for t 1 in equation ( 6 ) . 2 ( t 2 − 5 5 ) − 1 = − t 2 + 4 2 ( t 2 − 5 ) − 5 = − 5 t 2 + 20 7 t 2 = 35 t 2 = 5 Substitute t 2 − 5 5 for t 1 in equation ( 7 ) . 2 ( t 2 − 5 5 ) − 2 = t 2 − 7 2 ( t 2 − 5 ) − 10 = 5 t 2 − 35 3 t 2 = 15 t 2 = 5 Since the value of t 2 is same in both the above cases, the lines intersect To determine (b) Whether or not the given lines have a single point of intersection and if they do, determine an equation of a plane containing them.

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Question

Chapter 12.5, Problem 28E

To determine

(a)

Whether or not the given lines have a single point of intersection and if they do, determine an equation of a plane containing them.

To determine

(b)

Whether or not the given lines have a single point of intersection and if they do, determine an equation of a plane containing them.

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Chapter 12.5, Problem 27E

Chapter 12.5, Problem 29E

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Chapter 12 Solutions

Calculus: Early Transcendentals

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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(a) Whether or not the given lines have a single point of intersection and if they do, determine an equation of a plane containing them.

Solution Summary: The author explains the parametric equations of the lines passing through the point and parallel to the vector v=langle a,b,c.