Let $\overrightarrow{a}=\hat{i}+\hat{j}+2\hat{k},\overrightarrow{b}=2\hat{i}-3\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+\hat{k}$ 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 :

Since $\overrightarrow{v}$ in the plane of $\overrightarrow{a }and \overrightarrow{b}$,

$$\begin{gathered} \overrightarrow{v}=\lambda \overrightarrow{a}+\mu \overrightarrow{b} \\ \overrightarrow{v}=\lambda (1,1,2)+\mu (2,-3,1) \\ \overrightarrow{v}=(\lambda +2\mu ,\lambda -3\mu ,2\lambda +\mu ) \\ given that \overrightarrow{v}·\hat{j}=7 , \overrightarrow{v}·\frac{\overrightarrow{c}}{\left|\overrightarrow{c}\right|}=\frac{2}{\sqrt{3}} \\ \lambda -3\mu =7 \overrightarrow{V}·\overrightarrow{c}=2 \\ \lambda +2\mu -\lambda +3\mu +2\lambda +\mu =2 \\ 2\lambda +6\mu =2 \\ \lambda +3\mu =1 \\ \lambda -3\mu =7 \\ 2\lambda =8 \\ \lambda =4 \\ \mu =-1 \end{gathered}$$

we get $\overrightarrow{v}=(2,7,7)$