Let a =2 i + j -2 k, b = i + j and c = a × b. Let d be a vector such that | d - a |=√ 11,| c × d |=3 and the angle between c and d is (π)/4. Then a ·…

Let $\displaystyle \vec{a}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \vec{b}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $\displaystyle \vec{c}=\vec{a} \times \vec{b}$. Let $\displaystyle \vec{d}$ be a vector such that $\displaystyle |\vec{d}-\vec{a}|=\sqrt{11},|\vec{c} \times \vec{d}|=3$ and the angle between $\displaystyle \vec{c}$ and $\displaystyle \vec{d}$ is $\displaystyle \frac{\pi}{4}$. Then $\displaystyle \vec{a} \cdot \vec{d}$ is equal to