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 ) = ( cos t + t sin t ) i + ( sin t − t cos t ) j + 3 k . From Exercise 10 in Section 13.4, T = ( cos t ) i + ( sin t ) j and N = ( − sin t ) i + ( cos t ) j . Now obtain the binomial vector B by using the formula B = T × N as follows. B = T × N = | i j k cos t sin t 0 − sin t cos t 0 | = ( 0 ) i − ( 0 ) j + ( cos 2 t + sin 2 t ) k = ( cos 2 t + sin 2 t ) k = 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 = ( − sin t + t cos t + sin t ) i + ( cos t + t sin t − cos t ) j = ( t cos t ) i + ( t sin t ) j a = d v d t = ( − t sin t + cos t ) i + ( t cos t + sin t ) j Compute the vector | v × a | as follows

Thomas' Calculus: Early Transcendentals Plus Mymathlab With Pearson Etext -- Access Card Package (14th Edition) (hass, Heil & Weir, Thomas' Calculus Series)

14th Edition

ISBN: 9780134665573

Author: Joel R. Hass, Christopher D. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 13.5, Problem 10E

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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 9E

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Thomas' Calculus: Early Transcendentals Plus Mymathlab With Pearson Etext -- Access Card Package (14th Edition) (hass, Heil & Weir, Thomas' Calculus Series)

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