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.

4th Edition

ISBN: 9781319308858

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Concept explainers

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magn…

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.1, Problem 23E

Chapter 13.1, Problem 25E

Students have asked these similar questions

The line L is parallel to the plane x+y+z=2 and is perpendicular to the vector (-1,1,–2). If the line L contains the point (0, 1, 2), find the parametric equations for the line L.

Chapter 13 Solutions

CALCULUS:EARLY TRANS.(LL)-W/ACCESS

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

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

(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.