$$\begin{gathered} \mathrm{A} \mathrm{vector} \overline{\mathrm{a}} \mathrm{of} \mathrm{length} 2 \mathrm{units} \mathrm{is} \mathrm{making} \mathrm{an} \\ \mathrm{angle} {60}^0 \mathrm{with} \mathrm{each} \mathrm{of} \mathrm{the} \mathrm{X}-\mathrm{axis} \mathrm{and} \mathrm{Y}-\mathrm{axis}. \\ \mathrm{If} \mathrm{another} \mathrm{vector} \overline{\mathrm{b}} \mathrm{of} \mathrm{length} \sqrt{2} \mathrm{units} \mathrm{is} \mathrm{making} \\ \mathrm{an} \mathrm{angle} {45}^0 \mathrm{with} \mathrm{each} \mathrm{of} \mathrm{the} \mathrm{Y}-\mathrm{axis} \\ \mathrm{and} \mathrm{Z}-\mathrm{axis}, \mathrm{then} \overline{\mathrm{a}}\times \overline{\mathrm{b}}= \end{gathered}$$

$\begin{array}{rcl}{\cos }^2\mathrm{\alpha }+{\cos }^2 \mathrm{\beta }+{\cos }^2 \mathrm{\gamma }& =& 1\\ \mathrm{\alpha }& =& {60}^{°} \mathrm{\beta }={60}^{°} \Rightarrow {\cos }^2 \mathrm{\gamma }=1-\frac{1}{4}-\frac{1}{4}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ & \Rightarrow & \cos \mathrm{\gamma }=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.\Rightarrow \mathrm{v}={45}^{°}\\ \overline{\mathrm{a}}& =& 2(\frac{1}{2}\mathrm{i}+\frac{1}{2}\mathrm{j}+\frac{1}{\sqrt{2}}\mathrm{k})\\ {\cos }^2 \mathrm{\alpha }+{\cos }^2 \mathrm{\beta }+{\cos }^2 \mathrm{\gamma }& =& 1\\ \mathrm{\beta }& =& \mathrm{\gamma }={45}^{°} \Rightarrow {\cos }^2 \mathrm{\alpha }=1-\frac{1}{2}-\frac{1}{2}=0 \Rightarrow \mathrm{\alpha }={90}^{°}\\ \overline{\mathrm{b}}& =& {\mathrm{\gamma }}_2(0\mathrm{i}+\frac{1}{\sqrt{2}}\mathrm{j}+\frac{1}{\sqrt{2}}\mathrm{k})=\mathrm{j}+\mathrm{k}\\ \overline{\mathrm{a}}\times \overline{\mathrm{b}}& =& \left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 1& \sqrt{2}\\ 0& 1& 1\end{array}\right|=\mathrm{i}(1-\sqrt{2})-\mathrm{j}\left(\mathrm{i}\right)+\mathrm{k}\left(1\right)\end{array}$