Explanation Given: The parameterized curve r ( t ) = 〈 2 sin t , 2 cos t 〉 . Formula used: Suppose r is a smooth parameterized curve. The unit tangent vector T is r ′ | r ′ | . The principal unit normal vector is N ( t ) = T ′ ( t ) | T ′ ( t ) | . Calculation: Differentiate r ( t ) to compute r ′ ( t ) . r ′ ( t ) = 〈 ( 2 sin t ) ′ , ( 2 cos t ) ′ 〉 = 〈 2 cos t , − 2 sin t 〉 Use magnitude formula to obtain the value of | r ′ ( t ) | . | r ′ ( t ) | = | 〈 2 cos t , − 2 sin t 〉 | = ( 2 cos t ) 2 + ( − 2 sin t ) 2 = 4 cos 2 t + 4 sin 2 t = 4 ( cos 2 t + sin 2 t ) On further simplification, | r ′ ( t ) | = 4 ( 1 ) = 2 Use unit tangent formula to compute T ( t ) . T ( t ) = 〈 2 cos t , − 2 sin t 〉 2 = 〈 2 cos t 2 , − 2 sin t 2 〉 = 〈 cos t , − sin t 〉 Thus, the unit tangent vector T ( t ) is 〈 cos t , − sin t 〉 . Use magnitude formula to compute the value of | T ( t ) | . | T ( t ) | = | 〈 cos t , − sin t 〉 | = ( cos t ) 2 + ( − sin t ) 2 = cos 2 t + sin 2 t = 1 = 1 (1) Differentiate T ( t ) to compute T ′ ( t )

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Question

Chapter 11.9, Problem 27E

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To find: The unit tangent vector T and the principal unit normal vector N for the parameterized curve r(t)=〈2sint,2cost〉 and to verify |T|=|N|=1 and T⋅N=0 .

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Chapter 11.9, Problem 26E

Chapter 11.9, Problem 28E

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Chapter 11 Solutions

Calculus: Early Transcendentals (2nd Edition)

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The unit tangent vector T and the principal unit normal vector N for the parameterized curve r ( t ) = 〈 2 sin t , 2 cos t 〉 and to verify | T | = | N | = 1 and T ⋅ N = 0 .

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To find: The unit tangent vector T and the principal unit normal vector N for the parameterized curve r(t)=〈2sint,2cost〉 and to verify |T|=|N|=1 and T⋅N=0 .

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