Question

Consider the surface xyz = 45. How do you find the unit normal vector to the surface at the point (3, 5, 3) with positive first coordinate?

Answer 1

`hat n = 1/sqrt(59) ((5),(3),(5))`

Explanation:

the gradient will give you the normal vector

so for level surface `f(x,y,z) = x,y,z - 45`

`nabla f = vec n = ((yz),(xz),(xy))`

`= ((5*3),(3*3),(3*5)) = ((15),(9),(15))`

so `vec n = ((5),(3),(5))` is a normal vector

and `hat n = 1/sqrt(59) ((5),(3),(5))` is a normal unit vector