Let a =2 i - j + k and b =λ j +2 k, λ ∈ Z be two vectors. Let c = a × b and d be a vector of magnitude 2 in y z -plane. If | c |=√ 53, then the…

Let $\displaystyle \vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\displaystyle \vec{b}=\lambda \hat{j}+2 \hat{k}, \lambda \in \boldsymbol{Z}$ be two vectors. Let $\displaystyle \vec{c}=\vec{a} \times \vec{b}$ and $\displaystyle \vec{d}$ be a vector of magnitude $\displaystyle 2$ in $\displaystyle y z$-plane. If $\displaystyle |\overrightarrow{\mathrm{c}}|=\sqrt{53}$, then the maximum possible value of $\displaystyle (\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{d}})^2$ is equal to :