Answer
$2x+2y+z=6$ and $x=1+2t; y=1+2t; z=2+t$
Work Step by Step
Formula to calculate the vector equation is:$\nabla f(r_0) \cdot (r-r_0)=0$
As we know that the equation of tangent for $f( 1,1,2)=\lt 2,2,1 \gt$ is given by
$2(x-1)+2(y-1)+1(z-2)=0 \implies 2x-2+2y-2+z-2 =0 $
Thus, $2x+2y+z=6$
Now, the parametric equations can be written as: $r-r_0+\nabla f(r_0) t$ for $\nabla f( 1,1,2)=\lt 2,2,1 \gt$:
So, $x=1+2t; y=1+2t; z=2+t$
