Answer
$4x-y-5z=4$ and $x=2+4t; y=-1-t; z=1-5t$
Work Step by Step
The vector equation can be calculated as: $\nabla f(r_0) \cdot (r-r_0)=0$
The equation of the tangent for $\nabla f( 2,-1,1)=\lt 4,-1,-5 \gt$ is:
$4(x-2)-1(y+1)-5(z-1)=0$
or, $4x-8-y-1-5z+5 =0 \implies 4x-y-5z=4$
Now, the parametric equations can be written as: $r-r_0+\nabla f(r_0) t$ for $\nabla f( 2,-1,1)=\lt 4,-1,-5 \gt$:
Thus, $x=2+4t; y=-1-t; z=1-5t$
Hence, $4x-y-5z=4$ and $x=2+4t; y=-1-t; z=1-5t$
