Given the curve r(a)=<6−a^2,a^3+1,1−a> find all the points where the tangent vector on r(a)

Frankie Burnett 2022-11-21 Answered
The question is the following:
Given the curve r ( a ) =< 6 a 2 , a 3 + 1 , 1 a > and the plane x + y + z = π, find all the points where the tangent vector on r(a) is parallel to the plane.
I know finding the tangent vector is the first part of the problem. That would be T ( a ) = < 2 a , 3 a 2 , 1 > ( 2 a ) 2 + ( 3 a 2 ) 2 + ( 1 ) 2 . But beyond there I don't know how to draw a relationship between the line and plane.
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Answers (1)

luthersavage6lm
Answered 2022-11-22 Author has 22 answers
A vector ( α , β , γ ) is parallel to that plane if and only if α + β + γ = 0. Therefore,
T ( a )  is parallel to the plane 2 a + 3 a 2 1 = 0 a = 1  or  a = 1 3 .
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