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

$$\begin{gathered} \begin{array}{rl} & \overrightarrow{\mathrm{v}}=\mathrm{\lambda }\overrightarrow{\mathrm{a}}+\mathrm{\mu }\overrightarrow{\mathrm{b}}\\ & \overrightarrow{\mathrm{v}}=\mathrm{\lambda }(1,1,2)+\mathrm{\mu }(2,-3,1)\\ & \overrightarrow{\mathrm{v}}=(\mathrm{\lambda }+2\mathrm{\mu },\mathrm{\lambda }-3\mathrm{\mu },2\mathrm{\lambda }+\mathrm{\mu })\end{array} \\ Projection of \overrightarrow{\mathrm{v}}on\stackrel{ \to }{\mathrm{c}}= \overrightarrow{\mathrm{v}}\cdot \frac{\overrightarrow{\mathrm{c}}}{\left|\overrightarrow{\mathrm{c}}\right|}=\frac{2}{\sqrt{3}} \\ \begin{array}{ll} & \mathrm{\lambda }+2\mathrm{\mu }-\mathrm{\lambda }+3\mathrm{\mu }+2\mathrm{\lambda }+\mathrm{\mu }=2\\ & 2\mathrm{\lambda }+6\mathrm{\mu }=2\\ & \mathrm{\lambda }+3\mathrm{\mu }=1\\ \overrightarrow{ v}·\hat{j}=7\Rightarrow & \mathrm{\lambda }-3\mathrm{\mu }=7\\ & 2\mathrm{\lambda }=8\\ & \mathrm{\lambda }=4\\ & \mathrm{\mu }=-1\\ & \mathrm{We} \mathrm{get} \overrightarrow{\mathrm{v}}=(2,7,7)\\ now& \overrightarrow{v}·(\hat{i}+\hat{k})=2+7=9\end{array} \end{gathered}$$