Explanation Given: The vector, r ( t ) = 2 cos t i + 2 sin t j + t k and a coordinate, P = ( 1 , 3 , π 3 ) . Formula used: Unit tangent vector T ( t ) to the curve, r ( t ) is given by, T ( t ) = r ′ ( t ) ‖ r ′ ( t ) ‖ , such that r ' ( t ) ≠ 0 Calculation: Consider the vector, r ( t ) = 2 cos t i + 2 sin t j + t k . Now, use the formula, T ( t ) = r ′ ( t ) ‖ r ′ ( t ) ‖ to find the tangent vector to the curve r ( t ) = 2 cos t i + 2 sin t j + t k . First calculate, r ′ ( t ) . So, r ′ ( t ) = − 2 sin t i + 2 cos t j + k . Now calculate, ‖ r ' ( t ) ‖ . So, ‖ r ′ ( t ) ‖ = ( − 2 sin t ) 2 + ( 2 cos t ) 2 + ( 1 ) = 4 sin 2 t + 4 cos 2 t + 1 = 4 + 1 = 5 Now put, r ′ ( t ) = − 2 sin t i + 2 cos t j + k and ‖ r ' ( t ) ‖ in the formula, T ( t ) = r ′ ( t ) ‖ r ′ ( t ) ‖ . So, T ( t ) = − 2 sin t i + 2 cos t j + k 5 = − 2 5 sin t i + 2 5 cos t j + 1 5 k Now at P = ( 1 , 3 , π 3 ) , the value of t in the expression, r ( t ) = 2 cos t i + 2 sin t j + t k is t = π 3

10th Edition

ISBN: 9781285057095

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 12, Problem 45RE

To determine

To calculate: The unit tangent vector to the curve r(t)=2costi+2sintj+tk and set of parametric equations for the line tangent to the space curve at P=(1,3,π3).

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Chapter 12, Problem 44RE

Chapter 12, Problem 46RE

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The unit tangent vector to the curve r ( t ) = 2 cos t i + 2 sin t j + t k and set of parametric equations for the line tangent to the space curve at P = ( 1 , 3 , π 3 ) .

Solution Summary: The author calculates the unit tangent vector to the curve r(t)=2mathrmcosti+2

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To calculate: The unit tangent vector to the curve r(t)=2costi+2sintj+tk and set of parametric equations for the line tangent to the space curve at P=(1,3,π3).

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Chapter 12 Solutions