How do you find the unit normal vector to the surface defined by the equation xyz + x²y - 5yz - 5 = 0 at the point (3, 1, 2)?

To find the unit normal vector to the surface defined by the equation xyz + x²y - 5yz - 5 = 0 at the point (3, 1, 2), first compute the gradient of the function. The gradient at the point (3, 1, 2) is given as (8i + 5j - 2k). The unit normal vector is then obtained by normalizing this gradient vector.

The surface is defined by the scalar function f(x, y, z) = xyz + x²y - 5yz - 5. To find the unit normal vector, we calculate the gradient of this function, denoted as ∇f. At the point (3, 1, 2), the gradient is calculated to be (8, 5, -2). The magnitude of this gradient vector is √(8² + 5² + (-2)²) = √(64 + 25 + 4) = √93. The unit normal vector N is then found by dividing the gradient vector by its magnitude, resulting in N = (8/√93)i + (5/√93)j - (2/√93)k.