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 ) = ( 6 sin 2 t ) i + ( 6 cos 2 t ) j + 5 t k . From Exercise 12 in Section 13.4, T = ( 12 13 cos 2 t ) i − ( 12 13 sin 2 t ) j + 5 13 k and N = ( − sin 2 t ) i + ( − cos 2 t ) j . Now obtain the binomial vector B by using the formula B = T × N as follows. B = T × N = | i j k 12 13 cos 2 t − 12 13 sin 2 t 5 13 − sin 2 t − cos 2 t 0 | = ( 0 + 5 13 cos 2 t ) i − ( 0 + 5 13 sin 2 t ) j + ( − 12 13 cos 2 2 t − 12 13 sin 2 t ) k = ( 5 13 cos 2 t ) i − ( 5 13 sin 2 t ) j + ( − 12 13 ( cos 2 t + sin 2 t ) ) k = ( 5 13 cos 2 t ) i − ( 5 13 sin 2 t ) j + 12 13 k Therefore, the binomial vector for the given space curve is B = ( 5 13 cos 2 t ) i − ( 5 13 sin 2 t ) j + 12 13 k _ . Obtain the velocity as shown below. v = d r d t = ( 6 cos 2 t ) ( 2 ) i + ( − 6 sin 2 t ) ( 2 ) j + 5 k = ( 12 cos 2 t ) i + ( − 12 sin 2 t ) j + 5 k Compute the acceleration as shown below. a = d v d t = ( − 12 sin 2 t ) ( 2 ) i + ( − 12 cos 2 t ) ( 2 ) j = ( − 24 sin 2 t ) i + ( − 24 cos 2 t ) j Compute the vector | v × a | as follows

14th Edition

ISBN: 9780134438986

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

Publisher: PEARSON

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Chapter 13.5, Problem 12E

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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 11E

Chapter 13.5, Problem 13E

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