To find the unit normal vector to the surface at the point (1, 4, 5), we can use the gradient vector.
The gradient vector is a vector that points in the direction of the maximum rate of increase of a function. In this case, we can think of the equation xyz = 20 as a function f(x, y, z) = xyz - 20.
The gradient vector of a function is a vector whose components are the partial derivatives of the function with respect to each variable. So, to find the gradient vector of f(x, y, z) = xyz - 20, we need to find the partial derivatives.
∂f/∂x = yz
∂f/∂y = xz
∂f/∂z = xy
Evaluate these partial derivatives at the point (1, 4, 5):
∂f/∂x = (4)(5) = 20
∂f/∂y = (1)(5) = 5
∂f/∂z = (1)(4) = 4
The gradient vector at the point (1, 4, 5) is then:
∇f(1, 4, 5) = (20, 5, 4)
To find the unit normal vector, we need to normalize this vector by dividing each component by its magnitude.
Magnitude of ∇f(1, 4, 5) = sqrt((20)^2 + (5)^2 + (4)^2) = sqrt(541) ≈ 23.258
Unit normal vector = (∇f(1, 4, 5))/|∇f(1, 4, 5)| = (20/23.258, 5/23.258, 4/23.258)
≈ (0.86, 0.22, 0.17)
Therefore, the unit normal vector to the surface at the point (1, 4, 5) with positive first coordinate is (0.86, 0.22, 0.17).