Official solution
\begin{gathered} \begin{gathered} \quad \vec{a}=2\hat{i}+\hat{j}-2\hat{k},\vec{b}=\hat{i}-\hat{j} \\ |\vec{a}|=\sqrt{4+1+4}=3,|\vec{b}|=\sqrt{1+1}=\sqrt{2} \\ \vec{c}=\vec{a}\times \vec{b} \\ \vec{c}=|\begin{array}{|c|}\hat{i} \quad \hat{j} \quad \hat{k} \\ 2 \quad 1 \quad -2 \\ 1 \quad -1 \quad 0\end{array}| \\ \vec{c}=\hat{i}(0+2)-\hat{j}(0+2)+\hat{k}(-2-1) \\ \vec{c}=2\hat{i}-2\hat{j}-3\hat{k} \end{gathered}
\begin{gathered} |\vec{c}|=\sqrt{4+4+1}=\sqrt{9}=3 \\ |\vec{c}\times \vec{d}|=3\quad \vec{c}\quad \vec{d}\pi /4 \\ |\vec{c}||\vec{d}|\sin \frac{\pi }{4}=3 \\ 3|\vec{d}|\times \frac{1}{\sqrt{2}}=3\Rightarrow |\vec{d}|=\sqrt{2} \\ |\vec{d}-\vec{a}|=\sqrt{11}() \\ |\vec{d}-\vec{a}{|}^{2}=11 \\ |\vec{d}{|}^{2}+|\vec{a}{|}^{2}-2\vec{d}\cdot \vec{a}=11 \\ 2+9-2\vec{d}\cdot \vec{a}=11|\vec{a}|=3 \\ 11-2\vec{d}\cdot \vec{a}=11 \\ -2\vec{d}\cdot \vec{a}=0 \\ \vec{d}\cdot \vec{a}=0 \end{gathered} \end{gathered}