Exercise 42

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 have an equation for \(S\) but you know that the curves
\[ \mathbf{r}_1(t) = \langle 2 + 3t,\ 1 - t^2,\ 3 - 4t + t^2 \rangle \\ \mathbf{r}_2(u) = \langle 1 + u^2,\ 2u^3 - 1,\ 2u + 1 \rangle \]
both lie on \(S\). Find an equation of the tangent plane at \(P\).