Unit Normal Vector to Surface

To find the unit normal vector to a surface at a given point, we need to determine the gradient vector of the surface equation and then normalize it.

Given:
Surface equation: xy + yz + zx = 1
Point: (1, 1, 1)

Finding the Gradient Vector
The gradient vector of a surface equation is given by the partial derivatives of the equation with respect to each variable. In this case, we have the surface equation:

f(x, y, z) = xy + yz + zx - 1

Taking the partial derivatives:

∂f/∂x = y + z
∂f/∂y = x + z
∂f/∂z = y + x

Thus, the gradient vector is:

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) = (y + z, x + z, y + x)

Evaluating the Gradient Vector at the Given Point
To find the unit normal vector, we need to evaluate the gradient vector at the given point (1, 1, 1).

∇f(1, 1, 1) = (1 + 1, 1 + 1, 1 + 1) = (2, 2, 2)

Normalizing the Vector
To normalize the vector, we divide each component by the magnitude of the vector. The magnitude of the vector is given by:

|∇f(1, 1, 1)| = √(2^2 + 2^2 + 2^2) = √12 = 2√3

Therefore, the unit normal vector to the surface at the point (1, 1, 1) is:

n = (∇f(1, 1, 1)) / |∇f(1, 1, 1)| = (2/2√3, 2/2√3, 2/2√3) = (1/√3, 1/√3, 1/√3)

Explanation:
- We first find the gradient vector of the surface equation by taking the partial derivatives of the equation with respect to each variable.
- We then evaluate the gradient vector at the given point (1, 1, 1) to obtain the vector (2, 2, 2).
- To normalize the vector, we divide each component by the magnitude of the vector.
- The resulting unit normal vector to the surface at the point (1, 1, 1) is (1/√3, 1/√3, 1/√3).

Note: In the above explanation, HTML tags such as b and br have been used to format the text.