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## Homework Statement

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

r

_{1}(t)=<2+3t,1-t^2,3-4t+t^2>

r

_{2}(u)=<1+u^2,2u^3-1,2u+1>

both lie on S. Find an equation of the tangent plane at P.

## Homework Equations

z-z

_{0}=a(x-x

_{0})+b(y-y

_{0}) The equation for a plane where a=f

_{x}(x

_{0},y

_{0}) and b=f

_{y}(x

_{0},y

_{0})

## The Attempt at a Solution

I know that r

_{1}and r

_{2}intercept at P when u=1 and t=0, so I think x

_{0}=2, y

_{0}=1, and z

_{0}=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

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