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.

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ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

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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(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.