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 ) = ( cos 3 t ) i + ( sin 3 t ) j , 0 < t < π 2 (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 = ( 3 cos 2 t ( − sin t ) ) i + ( 3 sin 2 t ( cos t ) ) j = ( − 3 cos 2 t sin t ) i + ( 3 sin 2 t cos t ) j Substitute ( − 3 cos 2 t sin t ) i + ( 3 sin 2 t cos t ) j for d r ( t ) d t in equation (2) to find v . v = ( − 3 cos 2 t sin t ) i + ( 3 sin 2 t cos t ) j Take the magnitude of above equation to find | v | . | v | = | ( − 3 cos 2 t sin t ) i + ( 3 sin 2 t cos t ) j | = ( − 3 cos 2 t sin t ) 2 + ( 3 sin 2 t cos t ) 2 + 4 2 = 9 cos 4 t sin 2 t + 9 sin 4 t cos 2 t = 9 cos 2 t sin 2 t ( cos 2 t + sin 2 t ) Simplify the above equation to find | v | . | v | = 9 cos 2 t sin 2 t ( 1 ) { ∵ cos 2 θ + sin 2 θ = 1 } = 9 cos 2 t sin 2 t = 3 cos t sin t Substitute ( − 3 cos 2 t sin t ) i + ( 3 sin 2 t cos t ) j for v , and 3 cos t sin t for | v | in equation (3) to find T

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Chapter 13.4, Problem 14E

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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 13E

Chapter 13.4, Problem 15E

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