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 ) = ( t 3 3 ) i + ( t 2 2 ) j , t > 0 (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 t 2 3 ) i + ( 2 t 2 ) j = t 2 i + t j Substitute t 2 i + t j for d r ( t ) d t in equation (2) to find v . v = t 2 i + t j Take the magnitude of above equation to find | v | . | v | = | t 2 i + t j | = ( t 2 ) 2 + t 2 = t 4 + t 2 = t 2 ( t 2 + 1 ) | v | = t t 2 + 1 Substitute ( 3 cos t ) i − ( 3 sin t ) j + 4 k for v , and t t 2 + 1 for | v | in equation (3) to find T . T = t 2 i + t j t t 2 + 1 = t ( t i + j ) t t 2 + 1 = ( t i + j ) t 2 + 1 = t t 2 + 1 i + 1 t 2 + 1 j Differentiate above equation with respect to t . d T d t = [ ( 1 ) ( 1 + t 2 ) − ( t ) ( 1 2 ) ( 1 + t 2 ) 3 2 ( 2 t ) ( 1 + t 2 ) 2 ] i + [ ( 0 ) ( 1 + t 2 ) − ( 1 ) ( 1 2 ) ( 1 + t 2 ) 3 2 ( 2 t ) ( 1 + t 2 ) 2 ] j = [ ( 1 + t 2 ) − ( t 2 ) ( 1 + t 2 ) − 1 2 ( 1 + t 2 ) 2 ] i + [ − t ( 1 + t 2 ) − 1 2 ( 1 + t 2 ) 2 ] j = [ ( 1 + t 2 ) − ( t 2 ) ( 1 1 + t 2 ) ( 1 + t 2 ) 2 ] i + [ − t ( 1 1 + t 2 ) ( 1 + t 2 ) 2 ] j = [ 1 + t 2 ( 1 + t 2 ) 2 − ( t 2 ) ( 1 + t 2 ) 3 ] i + [ − t ( 1 + t 2 ) 3 ] j Simplify the above equation to find d T d t

4th Edition

ISBN: 9780134439099

Author: Hass, Joel., Heil, Christopher , WEIR, Maurice D.

Publisher: Pearson,

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Chapter 13.4, Problem 13E

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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 12E

Chapter 13.4, Problem 14E

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