To determine Find the unit tangent vector and the length of the indicated portion of the given curve. Explanation Given information: The smooth curve is, r ( t ) = ( t cos t ) i + ( t sin t ) j + 2 2 3 t 3 2 k , 0 ≤ t ≤ π Formula used: Write the expression to calculate the velocity vector. v = d r d t (1) Write the expression to calculate the unit tangent vector of the smooth curve. T = v | v | (2) Write the expression to calculate the arc length of the curve. L = ∫ a b | v | d t (3) Calculation: Substitute the given curve function in equation (1) to find v . v = d d t ( ( t cos t ) i + ( t sin t ) j + 2 2 3 t 3 2 k ) = ( cos t ( 1 ) + t ( − sin t ) ) i + ( sin t ( 1 ) + t ( cos t ) ) j + 2 2 3 ( 3 2 ) t 3 2 − 1 k = ( cos t − t sin t ) i + ( sin t + t cos t ) j + 2 t 1 2 k = ( cos t − t sin t ) i + ( sin t + t cos t ) j + 2 t k To find | v | : | v | = | ( cos t − t sin t ) i + ( sin t + t cos t ) j + 2 t k | = ( cos t − t sin t ) 2 + ( sin t + t cos t ) 2 + ( 2 t ) 2 = cos 2 t + t 2 sin 2 t − 2 t sin t cos t + sin 2 t + t 2 cos 2 t + 2 t sin t cos t + 2 t = ( cos 2 t + sin 2 t ) + t 2 ( sin 2 t + cos 2 t ) + 2 t Simplify the above equation to find | v |

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Author: Hass

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Chapter 13.3, Problem 7E

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Find the unit tangent vector and the length of the indicated portion of the given curve.

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Chapter 13.3, Problem 6E

Chapter 13.3, Problem 8E

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