Space Curve: Equation of normal plane at point...

Question

If I have the space curve $r(t) = \langle t, t^2, t^3 \rangle$, how would I find an equation of the normal plane to $r(t)$ at the point $P(2,4,8)$?

Answer 1

1. A plane containing the origin with normal vector $\mathbf{n}$ is given by $\mathbf{n}\cdot\mathbf{x}=0$ (this is simply a state-ment of orthogonality). As corollary, a plane containing the point $\mathbf{p}$ is given by $\mathbf{n}\cdot(\mathbf{x}-\mathbf{p})=0$.
2. A plane normal to a curve $\gamma$ at a point $\mathbf{r}(t)=\mathbf{p}$ is exactly the plane containing the point $\mathbf{p}$ with normal vector given by the curve's tangent vector at this very point, $\mathbf{n}=\mathbf{T}(t)=\mathbf{r}\,'(t)$.
3. For what $t$ is $\mathbf{r}(t)=(2,4,8)$? What is the derivative $\mathbf{r}\,'(t)$ evaluated at this particular time $t$?

Answer 2

The normal vector at the point P is (1,4,12) and the normal plane is given by $x+4y+12z=114$ using Hessian normal form of a plane.