Answer
a) $x+y+z=1$
b) $x=t,y=1+t; z=t$
Work Step by Step
a) Since, we have the vector equation $r(x,y,z)=r_0+t \nabla f(r_0)$
The equation of the tangent line is: $\nabla f(0,1,0)=\lt 1,1,1 \gt$
Thus, $1(x-0)+1(y-1)-1(z-0)=0$
or, $x+y+z=1$
b) Since, we have the vector equation $r=r_0+t \nabla f(r_0)$
Now, the parametric equations are:
$x=0+t=t,y=1+t; z=0+t=t$
