How am I supposed to find the equation of a tangent plane on a surface that its equation is not explicit defined in terms of z?
The equation of the surface is:
$$
x^{2} -y^{2} -z^{2} = 1
$$
And I need to find all the points where the tangent plane is parallel to the plane:
$$z = x + y
$$
How can I do that?
My approach at the moment was to think about the gradient vector of the function as being the normal vector of the plane… But the gradient vector of that surface has only two components (partial x and y), and the vector the normal vector of the plane has three components (1,1,-1)…
Best Answer
Don't consider the surface as the graph of one (or more) function(s) of the form $z=g(x,y)$, but as a level set of a function of three variables: $$f(x,y,z) = c$$ where in your case $f(x,y,z)=x^2-y^2-z^2$ and $c=1$. Then the gradient of $f$ is: $$\nabla f = \left( \tfrac{\partial f}{\partial x}, \tfrac{\partial f}{\partial y}, \tfrac{\partial f}{\partial z} \right) = (2x,-2y,-2z)$$and this gradient is perpendicular to the surface, so parallel to a tangent plane at that point.