  Marstrand's projection theorem from $1954$ states that if $K \subset
\mathbb{R}^{3}$ is an analytic set, then, for $\mathcal{H}^{2}$ almost every $e
\in S^{2}$, the orthogonal projection $\pi_{e}(K)$ of $K$ to the line spanned
by $e$ has Hausdorff dimension $\min\{\dim_{\mathrm{H}} K,1\}$. This paper
contains the following sharper version of Marstrand's theorem. Let $V \subset
\mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. Then, for
$\mathcal{H}^{1}$ almost every $e \in S^{2} \cap V$, the projection
$\pi_{e}(K)$ has Hausdorff dimension $\min\{\dim_{\mathrm{H}} K,1\}$. For $0
\leq t < \dim_{\mathrm{H}} K$, we also prove an upper bound for the Hausdorff
dimension of those vectors $e \in S^{2} \cap V$ with $\dim_{\mathrm{H}}
\rho_{e}(K) \leq t < \dim_{\mathrm{H}} K$.
