Explanation Given: r ( t ) = 〈 t 2 − 1 , t − 2 t 2 , 4 − 6 t 〉 Calculation: First we will find equation of plane. r ( t ) = 〈 t 2 − 1 , t − 2 t 2 , 4 − 6 t 〉 ⇒ x = t 2 − 1 Then, 2 x = 2 t 2 − 2..... ( i ) ⇒ y = t − 2 t 2 ..... ( i i ) ⇒ z = 4 − 6 t Then, z 6 = 4 6 − 6 6 t z 6 = 2 3 − t ..... ( i i i ) Adding 3 equations, we get ⇒ 2 x + y + z 6 = 2 t 2 − 2 + t − 2 t 2 + 2 3 − t ⇒ 2 x + y + z 6 = 2 3 − 2 ⇒ 2 x + y + z 6 = − 4 3 ⇒ 12 x + 6 y + z 6 = − 4 3 ⇒ 12 x + 6 y + z = − 8 Therefore, equation of plane is 12 x + 6 y + z = − 8. Now, we will find points on the curve. At , t = 0 r ( 0 ) = 〈 0 2 − 1 , 0 − 2 × 0 2 , 4 − 6 × 0 〉 = 〈 − 1 , 0 , 4 〉 At , t = 1 r ( 1 ) = 〈 1 2 − 1 , 1 − 2 × 1 2 , 4 − 6 × 1 〉 = 〈 0 , − 1 , − 2 〉 At , t = 2 r ( 2 ) = 〈 2 2 − 1 , 2 − 2 × 2 2 , 4 − 6 × 2 〉 = 〈 3 , − 6 , − 8 〉 To determine (b) Show that for all t, the points on curve satisfy the equation of the plane.

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

4th Edition

ISBN: 9781319312886

Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert

Publisher: W. H. Freeman

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Question

Chapter 13.1, Problem 24E

To determine

(a)

Show that the points on the curve r(t)=〈t2−1,t−2t2,4−6t〉 at t = 0,1, and 2 do not lie on a line, and find the equation of the plane that they determine.

To determine

(b)

Show that for all t, the points on curve satisfy the equation of the plane.

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Chapter 13.1, Problem 23E

Chapter 13.1, Problem 25E

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Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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(a) Show that the points on the curve r ( t ) = 〈 t 2 − 1 , t − 2 t 2 , 4 − 6 t 〉 at t = 0,1, and 2 do not lie on a line, and find the equation of the plane that they determine.

Solution Summary: The author explains that the points on the curve r(t)=langle t2-1,t-2

Concept explainers

(a)

Show that the points on the curve r(t)=〈t2−1,t−2t2,4−6t〉 at t = 0,1, and 2 do not lie on a line, and find the equation of the plane that they determine.

(b)

Show that for all t, the points on curve satisfy the equation of the plane.

Want to see the full answer?

Chapter 13 Solutions

(a) Show that the points on the curve r ( t ) = 〈 t 2 − 1 , t − 2 t 2 , 4 − 6 t 〉 at t = 0,1, and 2 do not lie on a line, and find the equation of the plane that they determine.

Solution Summary: The author explains that the points on the curve r(t)=langle t2-1,t-2

Concept explainers

(a) Show that the points on the curve r(t)=〈t2−1,t−2t2,4−6t〉 at t = 0,1, and 2 do not lie on a line, and find the equation of the plane that they determine.

(b) Show that for all t, the points on curve satisfy the equation of the plane.

Want to see the full answer?

Chapter 13 Solutions

(a)

Show that the points on the curve r(t)=〈t2−1,t−2t2,4−6t〉 at t = 0,1, and 2 do not lie on a line, and find the equation of the plane that they determine.

(b)

Show that for all t, the points on curve satisfy the equation of the plane.