To determine The binomial vector B and the torsion τ for the given space curve. Explanation Formula used: 1. Binomial vector : “The binomial vector is given by B = T × N , where T is the unit tangent vector and N is the principal unit normal vector.” 2. Torsion : “The torsion is given by the formula τ = − d B d s ⋅ N = | x ˙ y ˙ z ˙ x ¨ y ¨ z ¨ x ⃛ y ⃛ z ⃛ | | v × a | 2 .” Calculation: The given space curve is r ( t ) = ( e t cos t ) i + ( e t sin t ) j + 2 k . From Exercise 11 in Section 13.4, T = ( cos t − sin t 2 ) i + ( sin t + cos t 2 ) j and N = ( − cos t − sin t 2 ) i + ( − sin t + cos t 2 ) j . Now obtain the binomial vector B by using the formula B = T × N as follows. B = | i j k cos t − sin t 2 sin t + cos t 2 0 − cos t − sin t 2 − sin t + cos t 2 0 | = ( 0 ) i − ( 0 ) j + ( ( cos t − sin t 2 ) 2 + ( sin t + cos t 2 ) 2 ) k = ( cos 2 t − 2 cos t sin t + sin 2 t + sin 2 t + 2 cos t sin t + cos 2 t 2 ) k = 2 k 2 = k Therefore, the binomial vector for the given space curve is B = k _ . Obtain the velocity and acceleration as shown below. v = d r d t = ( e t cos t − e t sin t ) i + ( e t sin t + e t cos t ) j a = d v d t = ( e t cos t − e t sin t − e t sin t − e t cos t ) i + ( e t sin t + e t cos t + e t cos t − e t sin t ) j = ( − 2 e t sin t ) i + ( 2 e t cos t ) j Compute the vector | v × a | as follows

Thomas' Calculus - MyMathLab Integrated Review

14th Edition

ISBN: 9780134786223

Author: Hass

Publisher: PEARSON

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Chapter 13.5, Problem 11E

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The binomial vector B and the torsion τ for the given space curve.

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Chapter 13.5, Problem 10E

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Thomas' Calculus - MyMathLab Integrated Review

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