Let a = i + j + k, b and c = j - k be three vectors such that a × b = c and a · b =1. If the length of projection vector of the vector b on the…

Let $\displaystyle \vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}$ and $\displaystyle \vec{c}=\hat{j}-\hat{k}$ be three vectors such that $\displaystyle \vec{a} \times \vec{b}=\vec{c}$ and $\displaystyle \vec{a} \cdot \vec{b}=1$. If the length of projection vector of the vector $\displaystyle \vec{b}$ on the vector $\displaystyle \vec{a} \times \vec{c}$ is $\displaystyle l$, then the value of $\displaystyle 3 l^2$ is equal to \_\_\_\_.