Question

Find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector ∇F(x,y,z).] ln(y−z x)=0 Surface (1,4,3) Point

Answer 1

To find the unit normal vector at the point (1,4,3) for the surface defined by ln(y-z/x)=0, one must compute the gradient of the implicit function F, normalize this gradient vector, and evaluate it at the specified point.

Step-by-step explanation:

The student is asking to find a unit normal vector to a given surface at a specific point by normalizing the gradient vector ∇F(x,y,z). In order to find a unit normal vector to the surface at the point (1,4,3), we need to calculate the gradient of F, where F is the implicit function given by ln(y-z/x)=0 and then normalize the gradient at the given point.

To begin with, the gradient vector of F is obtained by computing the partial derivatives of F with respect to x, y, and z. Once the gradient vector is found, it is then normalized by dividing it by its magnitude to obtain the unit normal vector. Since the given function is equivalent to y-z/x=1, for simplicity, the gradient can be found for F(x,y,z) = y-z/x - 1 = 0. We differentiate with respect to x, y, and z, obtaining the components of the gradient vector, which are then evaluated at the point (1,4,3).

After finding these components, the magnitude of the gradient vector is calculated, and then this gradient vector is divided by its magnitude to obtain the unit normal vector at (1,4,3).

Answer 2

The unit normal vector to the surface at the point (1,4,3) for the given function ln(y - zₓ) = 0 is ⟨ -1/√26, 3/√26, 4/√26 ⟩.

Step-by-step explanation:

To find the unit normal vector to the surface at the given point, we first need to compute the gradient vector ∇F(x, y, z) for the function ln(y - zₓ) = 0. The gradient vector is a vector of partial derivatives, and in this case, it is given by ∇F = ⟨ -z/(x(y - z)), 1/(y - z), x/(y - z) ⟩.

Now, evaluate the gradient vector at the point (1, 4, 3):

∇F(1, 4, 3) = ⟨ -3, 1/(-1), 1 ⟩ = ⟨ -3, -1, 1 ⟩.

Next, normalize this gradient vector to obtain the unit normal vector. The normalization involves dividing each component of the vector by its magnitude. The magnitude of the gradient vector is given by √((-3)² + (-1)² + 1²) = √11. Therefore, the unit normal vector is ⟨ -3/√11, -1/√11, 1/√11 ⟩.

Finally, express this unit normal vector in a more simplified form by multiplying each component by √11/√11 to rationalize the denominator. This results in the final unit normal vector ⟨ -1/√26, 3/√26, 4/√26 ⟩.

In conclusion, the unit normal vector to the surface at the point (1, 4, 3) is ⟨ -1/√26, 3/√26, 4/√26 ⟩, providing a direction perpendicular to the surface at that specific point.