Let a = i + j + k, b =2 i +2 j + k and d = a × b. If c is a vector such that a · c =| c |, | c -2 a |^2=8 and the angle between d and c is (π)/4,…

Let $\displaystyle \vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\displaystyle \overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}$. If $\displaystyle \overrightarrow{\mathrm{c}}$ is a vector such that $\displaystyle \vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|$, $\displaystyle |\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8$ and the angle between $\displaystyle \overrightarrow{\mathrm{d}}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ is $\displaystyle \frac{\pi}{4}$, then $\displaystyle |10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2$ is equal to $\displaystyle \_\_\_\_$.