Find unit normal to surface

Question

Find a unit normal to the given surface at given point. xyz=8 at (1,1,8)

Answer 1

f(x,y) = z = 8/xy

 

∂f/∂x = -8/x²y  

 

at (1,1,8) this is -8/1*1*1 = -8

 

∂f/∂y = -8/y²x  = -8/1*1*1 = -8

 

So the normal vector = (-8,-8,-1)

 

to make the magnitude of this vector equal to one:

 

we divide by sqrt((-8)² + (-8)² + (1)²) = sqrt(129) = 11.36

 

so the unit normal is (-.704,-.704,-.088)