Let a = i + j +2 k, b =2 i -3 j + k and c = i - j + k be three given vectors. Let v be a vector in the plane of a and b whose projection on c is 2/(√…

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