Question

Find the equation of the normal line to the surface 2=xeʸcos(z) at the point (4,0,0).

Answer 1

To find the equation of the normal line to the surface, we first need to find the gradient vector of the surface at the given point. The gradient vector will be perpendicular to the tangent plane and hence will be the equation of the normal line.

Step-by-step explanation:

To find the equation of the normal line to the surface, we first need to find the gradient vector of the surface at the given point. The gradient vector will be perpendicular to the tangent plane and hence will be the equation of the normal line.

So, let's find the gradient vector:

1. Find the partial derivative of the surface function with respect to each variable: f(x,y,z) = xe^(y)cos(z)
2. Substitute the coordinates of the given point into the partial derivatives to get the gradient vector: (∂f/∂x, ∂f/∂y, ∂f/∂z) = (4 * e^(0)cos(0), 0 * e^(0)cos(0), -0 * e^(0)sin(0)) = (4, 0, 0)

Therefore, the equation of the normal line to the surface 2 = xe^(y)cos(z) at the point (4,0,0) is x/4 = y/0 = z/0.