Explanation Given: r ( t ) = 〈 cos t , cos 2 t , sin t 〉 Formula used: sin x = sin α ⇒ x = n π + ( − 1 ) n α Calculation: For xy -plane, z = 0 . sin t = 0 ⇒ sin t = sin 0 [ ∵ sin x = sin α ⇒ x = n π + ( − 1 ) n α ] ⇒ t = n π + ( − 1 ) n α ⇒ t = n π + ( − 1 ) n × 0 ⇒ t = n π n = 0 , 1 , 2 , 3..... ⇒ t = 0 , π , 2 π , 3 π .... So, different values of t give different point of the intersection of the curve with the xy-plane. That means, the curve intersects the xy-plane infinitely many times. Now, at t = 0 , r ( 0 ) = 〈 cos 0 , cos 2 × 0 , sin 0 〉 = 〈 1 , 1 , 0 〉

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ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

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Chapter 13.1, Problem 10E

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To show that the path of the curve r(t)=〈cost,cos2t,sint〉 intersects the xy -plane infinitely many times but the underlying space curve intersects the xy

-plane only twice.

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To show that the path of the curve r ( t ) = 〈 cos t , cos 2 t , sin t 〉 intersects the x y -plane infinitely many times but the underlying space curve intersects the x y -plane only twice.

Solution Summary: The author explains that different values of t give different point of intersection of the curve with the xy-plane.

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To show that the path of the curve r(t)=〈cost,cos2t,sint〉 intersects the xy -plane infinitely many times but the underlying space curve intersects the xy

-plane only twice.

Want to see the full answer?

Chapter 13 Solutions

To show that the path of the curve r ( t ) = 〈 cos t , cos 2 t , sin t 〉 intersects the x y -plane infinitely many times but the underlying space curve intersects the x y -plane only twice.

Solution Summary: The author explains that different values of t give different point of intersection of the curve with the xy-plane.

Concept explainers

Videos

To show that the path of the curve r(t)=〈cost,cos2t,sint〉 intersects the xy -plane infinitely many times but the underlying space curve intersects the xy -plane only twice.

Want to see the full answer?

Chapter 13 Solutions

To show that the path of the curve r(t)=〈cost,cos2t,sint〉 intersects the xy -plane infinitely many times but the underlying space curve intersects the xy

-plane only twice.