Explanation Given information: The vector is given by, r ( t ) = 〈 1 − cos ( 2 t ) , t + sin t , t 2 〉 Proof: Considering the y z plane, we set x = 0 and consider the vector as, r ( t ) = 〈 0 , t + sin t , t 2 〉 Then, the variable y is given as, y = t + sin t and z = t 2 where − ∞ < t < ∞ . For any value of t , the value of sin t variates between − 1 to 1 . − 1 ≤ sin t ≤ 1 t − 1 ≤ t + sin t ≤ t + 1 t − 1 ≤ y ≤ t + 1 Hence, for all − ∞ < t < ∞ , there exists y such that − ∞ < y < ∞ As well as y , the variable z also depends on t . For all − ∞ < t < ∞ , t 2 is continuous and greater than or equal zero. t 2 ≥ 0 z ≥ 0 Hence, we can conclude that for any value of t , the vector r ( t ) = 〈 0 , t + sin t , t 2 〉 is continuous on the y z plane

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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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 14.1, Problem 11E

To determine

To prove: That the vector r(t) intersects yz plane infinite times but does not cross trough it.

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Chapter 14.1, Problem 10E

Chapter 14.1, Problem 12E

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Chapter 14 Solutions

CALCULUS (CLOTH)

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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To prove: That the vector r ( t ) intersects y z plane infinite times but does not cross trough it.

Solution Summary: The author proves that the vector r(t) intersects the yz plane infinite times but does not cross trough it.

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To prove: That the vector r(t) intersects yz plane infinite times but does not cross trough it.

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Chapter 14 Solutions