The question is the following:Given the curve r ( a ) =&lt; 6 − a 2 , a 3 + 1 , 1 − a &gt; 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 ) = &lt; − 2 a , 3 a 2 , − 1 &gt; ( − 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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The question is the following:Given the curve   r  (  a  )  =&amp;lt;  6  −      a    2    ,      a    3    +  1  ,  1  −  a  &amp;gt; 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  )  =            &amp;lt;      −      2      a      ,      3              a        2            ,      −      1      &amp;gt;              (      −      2      a              )        2            +      (      3              a        2                    )        2            +      (      −      1              )        2            . But beyond there I don&#039;t know how to draw a relationship between the line and plane.

luthersavage6lm · Accepted Answer

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                .

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

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