I am having a hard time finishing this problem up:
Consider the surface
$4 x^{2} + 9 y^{2} + 4 z^{2} = 17$
and the point $P = \left( 1, 1, 1 \right)$ on this surface.
A) Find the outward unit normal vector to the surface at point P.
B) Find the equation of the tangent plane to the surface at the point P.
We know that $\nabla f$ at a point $P$ is perpendicular to the level curve of f that goes through P. That makes part A fairly simple. We simply solve for $\frac{\nabla f(P)}{||\nabla f(P)||}$.
$$\nabla f=8i+18j+8k$$
$$||\nabla f||= \sqrt{8^2+18^2+8^2}$$
So the unit normal vector $=\frac{1}{\sqrt{452}}(8i+18j+8k)$.
Now my question is about part B, i honestly don't even know where to begin with this one. Any help? Thanks!
Best Answer
To find the equation of the plane you need the normal vector $n$ (which you already found) to the plane and a point $P$ and the equation is given by