Find two unit vectors that are normal to $\sqrt{\frac{x+z}{y-1}}=z^2$ at $P(3,5,1)$.
Attempt: I have $f(x,y,z)=\sqrt{\frac{x+z}{y-1}}-z^2$. First I found the gradient : $\nabla f=
So the two unit vectors would be the positive and negative normalized gradients: $\vec{u_1}=\frac{<\frac{1}{8}, \frac{-1}{8}, \frac{-15}{8}>}{||<\frac{1}{8}, \frac{-1}{8}, \frac{-15}{8}>||}=\frac{8}{\sqrt{227}}<\frac{1}{8}, \frac{-1}{8}, \frac{-15}{8}>=\frac{1}{\sqrt{227}}<1,-1,-15>$
$\vec{u_2}=-\frac{1}{\sqrt{227}}<1,-1,-15>$
However, in the answer key the answer is $(\pm\frac{1}{\sqrt{365}}<1,-1,-19>$). I checked the derivatives multiple times and the gradient as well and I have no idea why my answer does not match the answer key. I do not think it is a typo, so help please.