1. We are asked to compute the normal vector and the tangent plane at a given point $P$ on a surface. 2. The normal vector to a surface defined by a function $f(x,y,z) = 0$ at a point $P$ is given by the gradient vector $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$ evaluated at $P$. 3. The tangent plane at point $P=(x_0,y_0,z_0)$ can be expressed as: $$ \nabla f(P) \cdot (x - x_0, y - y_0, z - z_0) = 0 $$ which expands to: $$ \frac{\partial f}{\partial x}(x_0,y_0,z_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0,y_0,z_0)(y - y_0) + \frac{\partial f}{\partial z}(x_0,y_0,z_0)(z - z_0) = 0 $$ 4. To proceed, we need the explicit function $f(x,y,z)$ defining the surface and the coordinates of point $P$. 5. Once $f$ and $P$ are known, compute the partial derivatives, evaluate them at $P$, and write the equation of the tangent plane. 6. The normal vector is simply the gradient vector evaluated at $P$. Please provide the function $f(x,y,z)$ and the point $P$ to continue with the calculations.