To determine Find the value of unit tangent vector ( T ) , the principal unit normal vector ( N ) and the curvature ( κ ) . Explanation Given information: The expression of plane curve is, r ( t ) = ( 6 sin 2 t ) i + ( 6 cos 2 t ) j + 5 t k (1) Formula used: Write a general expression to calculate constant vector v . v = d r d t (2) Here, d r d t is the value of change in plane curve. Write a general expression to calculate the unit tangent vector. T = v | v | (3) Write a general expression to calculate the principal unit normal vector. N = ( d T d t ) | d T d t | (4) Here, d T d t is the value of change in unit tangent vector. Write a general expression to calculate the curvature. κ = 1 | v | | d T d t | (5) Calculation: Differentiate equation (1) with respect to t . d r ( t ) d t = ( 6 ( 2 ) cos 2 t ) i + ( 6 ( 2 ) ( − sin 2 t ) ) j + 5 ( 1 ) k = ( 12 cos 2 t ) i − ( 12 sin 2 t ) j + 5 k Substitute ( 12 cos 2 t ) i − ( 12 sin 2 t ) j + 5 k for d r ( t ) d t in equation (2) to find v . v = ( 12 cos 2 t ) i − ( 12 sin 2 t ) j + 5 k Take the magnitude of above equation to find | v | . | v | = | ( 12 cos 2 t ) i − ( 12 sin 2 t ) j + 5 k | = ( 12 cos 2 t ) 2 + ( − 12 sin 2 t ) 2 + 5 2 = 144 cos 2 2 t + 144 sin 2 2 t + 25 = 144 ( cos 2 2 t + sin 2 2 t ) + 25 Simplify the above equation to find | v | . | v | = 144 ( 1 ) + 25 { ∵ cos 2 θ + sin 2 θ = 1 } = 169 = 13 Substitute ( 12 cos 2 t ) i − ( 12 sin 2 t ) j + 5 k for v , and 13 for | v | in equation (3) to find T . T = ( 12 cos 2 t ) i − ( 12 sin 2 t ) j + 5 k 13 = ( 12 cos 2 t ) 13 i − ( 12 sin 2 t ) 13 j + 5 13 k = ( 12 13 cos 2 t ) i − ( 12 13 sin 2 t ) j + 5 13 k Differentiate above equation with respect to t

14th Edition

ISBN: 9780134438986

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

Publisher: PEARSON

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Chapter 13.4, Problem 12E

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Find the value of unit tangent vector (T), the principal unit normal vector (N) and the curvature (κ).

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Chapter 13.4, Problem 11E

Chapter 13.4, Problem 13E

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Chapter 13 Solutions

Thomas' Calculus (14th Edition)

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 - In Exercises 5–8, r(t) is the position of a...Ch. 13.1 - In Exercises 5–8, r(t) is the position of a...Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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