Let $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}, \vec{b}=2 \hat{i}-3 \hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$ be three given vectors. Let $\vec{v}$ be a vector in the plane of $\bar{a}$ and $\overrightarrow{\mathrm{b}}$ whose projection on $\overrightarrow{\mathrm{c}}$ is $\frac{2}{\sqrt{3}}$. If $\overrightarrow{\mathrm{v}} \cdot \hat{\mathrm{j}}=7$, then $\overrightarrow{\mathrm{v}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{k}})$ is equal to :