  Let $\gamma:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in
$\mathbb{R}^3$, that is to say,
$\det\big(\gamma(\theta),\gamma'(\theta),\gamma"(\theta)\big)\neq 0$. For each
$\theta\in[0,1]$, let $V_\theta=\gamma(\theta)^\perp$ and let
$\pi_\theta:\mathbb{R}^3\rightarrow V_\theta$ be the orthogonal projections. We
prove that if $A\subset \mathbb{R}^3$ is a Borel set, then for a.e. $\theta\in
[0,1]$ we have $\text{dim}(\pi_\theta(A))=\min\{2,\text{dim} A\}$. More
generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and
$0\le s\le 2$, define $E_s(A):=\{\theta\in[0,1]:
\text{dim}(\pi_\theta(A))<s\}$. We have $\text{dim}(E_s(A))\le
1+s-\text{dim}(A)$. We also prove that if $\text{dim}(A)>2$, then for a.e.
$\theta\in[0,1]$ we have $\mathcal{H}^2(\pi_\theta (A))>0$.
