Answer to Question #270308 in Calculus for JaytheCreator

Question

Find the unit normal to the surface 𝑦 = 𝑥 + 𝑧 3 at the point (1,2,1). 

Answer 1

$F(x, y, z)=x-y+z^3=0$

$F_x=1, F_y=-1, F_z=3z^2$

Point $(1, 2, 1)$

$F_x=1, F_y=-1, F_z=3(1)^2=3$

$\nabla F=i-j+3k$

$|\nabla F|=\sqrt{(1)^2+(-1)^2+(3)^2}=\sqrt{11}$

$n=\dfrac{\nabla F}{|\nabla F|}=\dfrac{1}{\sqrt{11}}i-\dfrac{1}{\sqrt{11}}j+\dfrac{3}{\sqrt{11}}k$