Explanation Given: The vector valued function r ( t ) = 2 cos t i + 2 sin t j + t 2 k and the value of t as t = π 2 . Formula used: For a smooth curve C , represented by r on an open interval I , the unit tangent vector T ( t ) at t is defined as T ( t ) = r ′ ( t ) ‖ r ′ ( t ) ‖ , r ′ ( t ) ≠ 0 …… (1) Where, r ′ ( t ) is first derivative of r ( t ) and ‖ r ′ ( t ) ‖ is the modulus of r ′ ( t ) at t . Also, if T ′ ( t ) ≠ 0 , then the principal unit normal vector N ( t ) at t is defined as N ( t ) = T ′ ( t ) ‖ T ′ ( t ) ‖ …… (2) Where, T ′ ( t ) is first derivative of T ( t ) and ‖ T ′ ( t ) ‖ is the modulus of T ′ ( t ) at t . Now, the binormal vector B ( t ) at t is defined as B ( t ) = T ( t ) × N ( t ) …… (3) Calculation: For the given vector valued function r ( t ) = 2 cos t i + 2 sin t j + t 2 k , take first derivative of the function as r ( t ) = 2 cos t i + 2 sin t j + t 2 k r ′ ( t ) = − 2 sin t i + 2 cos t j + 1 2 k Now, evaluate ‖ r ′ ( t ) ‖ at t = π 2 as, ‖ r ′ ( π 2 ) ‖ = ( − 2 sin ( π 2 ) ) 2 + ( 2 cos ( π 2 ) ) 2 + ( 1 2 ) 2 = 4 + 0 + 1 4 = 17 2 Since r ′ ( t ) ≠ 0 , substitute the values of r ′ ( t ) and ‖ r ′ ( t ) ‖ in equation (1) to get the tangent vector as, T ( t ) = − 2 sin t i + 2 cos t j + 1 2 k 17 2 = 2 17 17 ( − 2 sin t i + 2 cos t j + 1 2 k ) At t = π 2 , T ( t ) becomes, T ( π 2 ) = 2 17 17 ( − 2 sin ( π 2 ) i + 2 cos ( π 2 ) j + 1 2 k ) = 2 17 17 ( − 2 i + 0 j + 1 2 k ) = 17 17 ( − 4 i + k ) The tangent vector for the vector valued function r ( t ) = 2 cos t i + 2 sin t j + t 2 k at t = π 2 is T ( π 2 ) = 17 17 ( − 4 i + k )

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Chapter 12.4, Problem 51E

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To calculate: The tangent vector T, normal vector N and the binormal vector B=T×N for the vector valued function r(t)=2costi+2sintj+t2k at t=π2.

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Chapter 12.4, Problem 50E

Chapter 12.4, Problem 52E

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The tangent vector T , normal vector N and the binormal vector B = T × N for the vector valued function r ( t ) = 2 cos t i + 2 sin t j + t 2 k at t = π 2 .

Solution Summary: The author calculates the tangent vector T, normal vector N and binormal vector B=Ttimes N for the vector valued function r(t)=2mathrm

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To calculate: The tangent vector T, normal vector N and the binormal vector B=T×N for the vector valued function r(t)=2costi+2sintj+t2k at t=π2.

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Chapter 12 Solutions