Find the equation of the plane?

Question

If the vector function r(t)=<t+2, 2t^2-t+1, 3+t-t^3> is at the point (2, 1, 3) tangent to the plane containing the vector function s(u)=<u^2+2, -u+1, 2u^2-u+3>, find the equation of the plane.

Answer 1

The normal direction of the plane

= r'(0) X s(0)

= <1, -1, 1> X <2, 1, 3>

= <-4, -1, 3>

So, the equation of the plane is 4(x-2) + (y-1) - 3(z-3) = 0.