Explanation The angle of intersection of two curves is the angle between the two tangent vectors to the curves. As the intersection point is at origin, the value of scalar parameter t is 0. Formula: Write the expression to find the angle between two tangent vectors r ′ 1 ( t ) and r ′ 2 ( t ) at origin. θ = cos − 1 ( r ′ 1 ( 0 ) ⋅ r ′ 2 ( 0 ) | r ′ 1 ( 0 ) | | r ′ 2 ( 0 ) | ) (1) Here, r ′ 1 ( 0 ) is the tangent vector of curve r 1 ( t ) , which is derivative of vector r 1 ( t ) at t = 0 and r ′ 2 ( 0 ) is the tangent vector of curve r 2 ( t ) , which is derivative of vector r 2 ( t ) at t = 0 . Calculation of tangent vector r ′ 1 ( t ) : To find the derivative of the vector function, differentiate each component of the vector function. Differentiate each component of the vector function r 1 ( t ) = 〈 t , t 2 , t 3 〉 as follows. d d t [ r 1 ( t ) ] = 〈 d d t ( t ) , d d t ( t 2 ) , d d t ( t 3 ) 〉 Use the following formula to compute the expression. d d t t n = n t n − 1 Compute the expression d d t [ r 1 ( t ) ] = 〈 d d t ( t ) , d d t ( t 2 ) , d d t ( t 3 ) 〉 by using the formula as follows. r ′ 1 ( t ) = 〈 1 , 2 t , 3 t 2 〉 (2) Calculation of tangent vector r ′ 1 ( t ) at the intersection point of origin: Substitute 0 for t in equation (2), r ′ 1 ( 0 ) = 〈 1 , 2 ( 0 ) , 3 ( 0 ) 2 〉 = 〈 1 , 0 , 0 〉 Calculation of tangent vector r ′ 2 ( t ) : Differentiate each component of the vector function r 2 ( t ) = 〈 sin t , sin 2 t , t 〉 as follows. d d t [ r 2 ( t ) ] = 〈 d d t ( sin t ) , d d t ( sin 2 t ) , d d t ( t ) 〉 Use the following formulae to compute the expression. d d t sin t = cos t d d t sin n t = n cos n t Compute the expression d d t [ r 2 ( t ) ] = 〈 d d t ( sin t ) , d d t ( sin 2 t ) , d d t ( t ) 〉 by using the formulae as follows

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Question

Chapter 13.2, Problem 33E

To determine

To find: The angle of intersection of curves r1(t)=〈t, t2, t3〉 and r2(t)=〈sint, sin2t, t〉 at the origin.

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Chapter 13 Solutions

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The angle of intersection of curves r 1 ( t ) = 〈 t , t 2 , t 3 〉 and r 2 ( t ) = 〈 sin t , sin 2 t , t 〉 at the origin.

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To find: The angle of intersection of curves r1(t)=〈t, t2, t3〉 and r2(t)=〈sint, sin2t, t〉 at the origin.

Chapter 13 Solutions