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

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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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Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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