I am new to the site and I have been struggling through two of my homework problems and I have no clue how to do them.
The first question is: "Are there any points on surface $x^2 -y^2 -z^2 = 1$ where the tangent plane is parallel to the plane $z= x + y$?." I know you have to find the directional vector and you need to solve for the normal vector to the tangent plane that is shown by the gradient of function f. Besides doing the derivatives in terms of variables x, y, and z for function f I do not know where to go from there. What is the next step in this question?
For the upper half $z = \sqrt{x^2-y^2-1}$: $$ \frac{\partial z}{\partial x} = \frac{x}{\sqrt{x^2-y^2-1}},\qquad \frac{\partial z}{\partial x} = \frac{-y}{\sqrt{x^2-y^2-1}}. $$ For all the planes $z = x + y + c$ the corresponding partial derivatives are 1,1 and you must solve the system $$ \frac{x}{\sqrt{x^2-y^2-1}} = 1,\qquad \frac{-y}{\sqrt{x^2-y^2-1}} = 1. $$ Can you continue?