Explanation Given: The parameterized curve is r ( t ) = 〈 2 cos t , 2 sin t , − t 〉 . Formula used: Suppose T and N are the unit tangent vector and the principal unit normal vector for the smooth curve C . The unit binormal vector is given by B = T × N . The torsion is given by λ = − d B d s ⋅ N . 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 cos t ) ′ , ( 2 sin t ) ′ , ( − t ) ′ 〉 = 〈 2 ( − sin t ) , 2 cos t , − 1 〉 = 〈 − 2 sin t , 2 cos t , − 1 〉 Use magnitude formula to obtain the value of | r ′ ( t ) | . | r ′ ( t ) | = | 〈 − 2 sin t , 2 cos t , − 1 〉 | = ( − 2 sin t ) 2 + ( 2 cos t ) 2 + ( − 1 ) 2 = 4 sin 2 t + 4 cos 2 t + 1 = 4 ( sin 2 t + cos 2 t ) + 1 On further simplification, | r ′ ( t ) | = 4 ( 1 ) + 1 = 4 + 1 = 5 Use unit tangent formula to compute T ( t ) . T ( t ) = 〈 − 2 sin t , 2 cos t , − 1 〉 5 = 1 5 〈 − 2 sin t , 2 cos t , − 1 〉 Thus, the unit tangent vector T ( t ) is 1 5 〈 − 2 sin t , 2 cos t , − 1 〉 . Differentiate T ( t ) to compute T ′ ( t ) . T ′ ( t ) = 1 5 〈 ( − 2 sin t ) ′ , ( 2 cos t ) ′ , ( − 1 ) ′ 〉 = 1 5 〈 − 2 cos t , 2 ( − sin t ) , 0 〉 = 1 5 〈 − 2 cos t , − 2 sin t , 0 〉 = 2 5 〈 − cos t , − sin t , 0 〉 Use magnitude formula to obtain the value of | T ′ ( t ) | . | T ′ ( t ) | = | 1 5 〈 − 2 cos t , − 2 sin t , 0 〉 | = 1 5 ( − 2 cos t ) 2 + ( − 2 sin t ) 2 + 0 2 = 1 5 4 cos 2 t + 4 sin 2 t = 1 5 4 ( cos 2 t + sin 2 t ) On further simplification, | T ′ ( t ) | = 1 5 4 ( 1 ) = 2 5 Use principal unit normal formula to compute the value of N ( t ) . N ( t ) = 2 5 〈 − cos t , − sin t , 0 〉 2 5 = 〈 − cos t , − sin t , 0 〉 Thus, the principal unit normal vector N ( t ) for the curve r ( t ) is 〈 − cos t , − sin t , 0 〉

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Author: Briggs

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Chapter 14.5, Problem 45E

To determine

To compute: The unit binormal vector and the torsion for the parameterized curve r(t)=〈2cost,2sint,−t〉 .

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Chapter 14.5, Problem 44E

Chapter 14.5, Problem 46E

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The unit binormal vector and the torsion for the parameterized curve r ( t ) = 〈 2 cos t , 2 sin t , − t 〉 .

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To compute: The unit binormal vector and the torsion for the parameterized curve r(t)=〈2cost,2sint,−t〉 .

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