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