Question

Find the unit vector normal to the surface x^2 y^2 z^2

Answer 1

Answer:

{−1√14,3√14,√27}

Explanation:

Calling

f(x,y,z)=x3+y3+3xyz−3=0

The gradient of f(x,y,z) at point x,y,z is a vector normal to the surface at this point.

The gradient is obtained as follows

∇f(x,y,z)=(fx,fy,fz)=3(x2+yz,y2+xz,xy)at point
(1,2,−1) has the value
3(−1,3,2) and the unit vector is
{−1,3,2}√1+32+22={−1√14,3√14,√27}