I have a function, $w=x^2+y^2-z^2$, and its gradient vector, $\nabla w=(2x, 2y, -2y)$. How can I write the equation for its tangent plane? Is something like the following accurate? $$ p=\nabla w_\hat i(x,y,z)x +\nabla w_\hat j(x,y,z)y + \nabla w_\hat k(x,y,z)z$$
What is the convention for something like this?
Or you can think that if a plane is given by $$\pi: ax+by+cz = d$$ where $(a,b,c)$ is a vector normal to the plane, and remember that the gradient is normal to the level surface of the function, then the tangent plane at $(x_0, y_0, z_0)$ is: $$\frac{\partial w}{\partial x}(x_0,y_0,z_0) (x - x_0) + \frac{\partial w}{\partial y}(x_0, y_0, z_0) (y - y_0) + \frac{\partial w}{\partial z}(x_0, y_0, z_0) (z - z_0) = 0$$
The tangent plane is characterized by $$\nabla w(x_0,y_0,z_0) \cdot (x-x_0, y-y_0, z-z_0) = 0$$ for all $(x_0,y_0,z_0)$ in the surface such that $\nabla w(x_0,y_0,z_0) \neq (0,0,0)$.
