Explanation Given information: The equations r 1 ( t ) = 〈 t , 2 t − 1 , t − 3 〉 and r 2 ( t ) = 〈 3 t − 3 , 6 t − 1 , 3 t − 1 〉 . 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 are parallel if the direction vectors are scalar multiples of 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 ) = 〈 t , 2 t − 1 , t − 3 〉 . Then, x = t y = 2 t − 1 z = t − 3 } ,.... ( 3 ) Compare equations ( 1 ) and ( 3 ) . Here, P 1 = ( x 1 , y 1 , z 1 ) = ( 0 , − 1 , − 3 ) and v 1 = 〈 1 , 2 , 1 〉 . Use similar argument for r 2 ( t ) = 〈 3 t − 3 , 6 t − 1 , 3 t − 1 〉 . x = 3 t − 3 y = 6 t − 1 z = 3 t − 1 } ,.... ( 4 ) Compare equations ( 1 ) and ( 4 ) . Here, P 2 = ( x 2 , y 2 , z 2 ) = ( − 3 , − 1 , − 1 ) and v 2 = 〈 3 , 6 , 3 〉 . Observe that v 2 = 3 v 1 . By definition, r 1 is parallel to r 2 To determine (b) Whether or not the given lines are distinct parallel lines and if they are, 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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Chapter 12.5, Problem 31E

To determine

(a)

Whether or not the given lines are distinct parallel lines and if they are, determine an equation of a plane containing them.

To determine

(b)

Whether or not the given lines are distinct parallel lines and if they are, determine an equation of a plane containing them.

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Chapter 12.5, Problem 30E

Chapter 12.5, Problem 32E

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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 are distinct parallel lines and if they are, determine an equation of a plane containing them.

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