Answer
a) $3x+5y+4z=18$ and
b) $x=3+6t,y=5+10t; z=-4+8t$
Work Step by Step
a. As we know that 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: $\nabla f(3,5,-4)=\lt 6,10,8 \gt$
Thus, $6(x-3)+10(y-5)+8(z+4)=0$
or, $6x+10y+8z=6 \implies 3x+5y+4z=18$
b. As we know that the vector equation is given by:$r(x,y,z)=r_0+t \nabla f(r_0)$
The parametric equations for $\nabla f(3,5,-4)=\lt 6,10,8 \gt$
are:
$x=3+6t,y=5+10t; z=-4+8t$
