Final answer:
To find the parametric equations of the normal line to a given surface at a specified point, first find the components of the normal vector by evaluating the partial derivatives of the surface equation at the given point. The parametric equations of the normal line passing through the point can be written as x = x0 + t*Nx, y = y0 + t*Ny, and z = z0 + t*Nz.
Step-by-step explanation:
The parametric equations of the normal line to a surface at a specified point can be found by determining the normal vector to the surface at that point. To find the normal vector, we need to take the partial derivatives of the surface equation with respect to x, y, and z, and evaluate them at the given point. Let's assume the surface equation is given as f(x, y, z) = 0.
1. Find the partial derivatives ∂f/∂x, ∂f/∂y, and ∂f/∂z.
2. Evaluate these partial derivatives at the given point (x0, y0, z0) to find the components of the normal vector N = (∂f/∂x, ∂f/∂y, ∂f/∂z).
3. The parametric equations of the normal line passing through the point (x0, y0, z0) are x = x0 + t*Nx, y = y0 + t*Ny, and z = z0 + t*Nz, where Nx, Ny, and Nz are the components of the normal vector N.