= Solution
A two-dimensional <embedded submanifold> is a subset which near every point can be straightened by a smooth local coordinate change in the ambient space to an open piece of the coordinate plane $\mathbb R^2\times\{0\}$. Equivalently locally it has a regular smooth parametrization of rank two, with the induced local topology.
A <tangent vector> at $x$ is the <derivative> $\gamma'(0)$ of a smooth curve in the <submanifold> with $\gamma(0)=x$. These <derivatives> form the two-dimensional <tangent space> $T_xM$. A <normal vector> is a vector <orthogonal> to every <tangent vector>. Its space is $(T_xM)^\perp$, which has $\boxed{\dim (T_xM)^\perp=1}$ in the three-dimensional ambient Euclidean space.
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