Let a = i + j + k, b =3 i +2 j - k, c =λ j +μ k and d be a unit vector such that a × d = b × d and c · d =1. If c is perpendicular to a, then |3 λ d…

Let $\displaystyle \vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}-\hat{k}, \vec{c}=\lambda \hat{j}+\mu \hat{k}$ and $\displaystyle \hat{d}$ be a unit vector such that $\displaystyle \vec{a} \times \hat{d}=\vec{b} \times \hat{d}$ and $\displaystyle \vec{c} \cdot \hat{d}=1$. If $\displaystyle \vec{c}$ is perpendicular to $\displaystyle \vec{a}$, then $\displaystyle |3 \lambda \hat{d}+\mu \vec{c}|^2$ is equal to $\displaystyle \_\_\_\_$