Suppose you need to know an equation of the tangent plane to a surface S at the point P(2,1,3). You don't know the equation for S but you know that the curves
both lie on S. Find an equation of the tangent plane at P.
z-z0=a(x-x0)+b(y-y0) The equation for a plane where a=fx(x0,y0) and b=fy(x0,y0)
The Attempt at a Solution
I know that r1 and r2 intercept at P when u=1 and t=0, so I think x0=2, y0=1, and z0=3 but I'm not sure how to find the partial derivatives with respect to x and y of the plane. I tried taking dz/dt divided by dy/dt and comparing it with dz/du divided by dy/du but I got different answers for the two intercepting curves so I don't know what to do. Do I need to find an orthogonal vector- if so, how do I do that :P