Answer
$x=\dfrac{1}{2}-t; y=1; z=\dfrac{1}{2}+t$
Work Step by Step
The vector equation can be calculated as: $\nabla f(r_0) \cdot (r-r_0)=0$
and $\nabla f \times \nabla g=-i+k$
Now, the parametric equations can be written as: $r-r_0+\nabla f(r_0) t$ for $\nabla f( \dfrac{1}{2},1,\dfrac{1}{2})=\lt -1,0,1 \gt$:
Thus, $x=\dfrac{1}{2}-t; y=1+0t=1; z=\dfrac{1}{2}+t$
or, $x=\dfrac{1}{2}-t; y=1; z=\dfrac{1}{2}+t$