Solution
= Solution
Define $F(x)=x_1^2/a^2+x_2^2/b^2+x_3^2/c^2-1$. Its <gradient> is $2(x_1/a^2,x_2/b^2,x_3/c^2)$. It cannot vanish on $F^{-1}(0)$, because the only possible zero is the origin and that point has $F=-1$. The regular level-set theorem therefore makes the ellipsoid a two-dimensional <embedded submanifold>. Its <normal space> is
$$
\boxed{N_xM=\mathbb R\,(x_1/a^2,x_2/b^2,x_3/c^2),}
$$
and the <tangent space> is the <orthogonal> complement of that line.