Answer
$x+y+2z=3$
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 $\nabla f( 1,1,\dfrac{1}{2})=\lt -\dfrac{1}{2},-\dfrac{1}{2},-1 \gt$ is:
$-\dfrac{1}{2}(x-1)-\dfrac{1}{2}(y-1)-1(z-\dfrac{1}{2})=0$
or, $-\dfrac{1}{2}x-\dfrac{1}{2}y-z=\dfrac{-3}{2}$
Hence, $x+y+2z=3$
