Explanation Given: The parameterized curve r ( t ) = 〈 4 sin t , 4 cos t , 10 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 ) = 〈 ( 4 sin t ) ′ , ( 4 cos t ) ′ , ( 10 t ) ′ 〉 = 〈 4 cos t , − 4 sin t , 10 〉 Use magnitude formula to obtain the value of | r ′ ( t ) | . | r ′ ( t ) | = | 〈 4 cos t , − 4 sin t , 10 〉 | = ( 4 cos t ) 2 + ( − 4 sin t ) 2 + 10 2 = 16 cos 2 t + 16 sin 2 t + 100 = 16 ( cos 2 t + sin 2 t ) + 100 On further simplification, | r ′ ( t ) | = 16 ( 1 ) + 100 = 116 = 2 29 Use unit tangent formula to compute T ( t ) . T ( t ) = 〈 4 cos t , − 4 sin t , 10 〉 2 29 = 1 29 〈 4 cos t 2 , − 4 sin t 2 , 10 2 〉 = 1 29 〈 2 cos t , − 2 sin t , 5 〉 Thus, the unit tangent vector T ( t ) is 1 29 〈 2 cos t , − 2 sin t , 5 〉 . Use magnitude formula to compute the value of | T ( t ) | . | T ( t ) | = | 1 29 〈 2 cos t , − 2 sin t , 5 〉 | = 1 29 ( 2 cos t ) 2 + ( − 2 sin t ) 2 + 5 2 = 1 29 4 cos 2 t + 4 sin 2 t + 25 = 1 29 4 ( cos 2 t + sin 2 t ) + 25 On further simplification, | T ( t ) | = 1 29 4 ( 1 ) + 25 = 1 29 29 = 1 Thus, the value of | T ( t ) | (1) Differentiate T ( t ) to compute T ′ ( t ) . T ′ ( t ) = 1 29 〈 ( 2 cos t ) ′ , ( − 2 sin t ) ′ , 5 ′ 〉 = 1 29 〈 − 2 sin t , − 2 cos t , 0 〉 Use magnitude formula to obtain the value of | T ′ ( t ) |

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ISBN: 9780135182543

Author: Briggs

Publisher: PEARSON

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Question

Chapter 14.5, Problem 28E

To determine

To find: The unit tangent vector T and the principal unit normal vector N for the parameterized curve r(t)=〈4sint,4cost,10t〉 and to verify |T|=|N|=1 and T⋅N=0 .

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The unit tangent vector T and the principal unit normal vector N for the parameterized curve r ( t ) = 〈 4 sin t , 4 cos t , 10 t 〉 and to verify | T | = | N | = 1 and T ⋅ N = 0 .

Solution Summary: The author explains the unit tangent vector T and the principal unit normal vector N for the parameterized curve r(t)=langle 4mathrmsint,

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To find: The unit tangent vector T and the principal unit normal vector N for the parameterized curve r(t)=〈4sint,4cost,10t〉 and to verify |T|=|N|=1 and T⋅N=0 .

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