Mathematics - Vector Question with Solution | TestHub

$\begin{array}{rl}& |(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}})|=|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}||\overrightarrow{\mathrm{c}}|\frac{\sqrt{3}}{2}\\ & |\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=2\sqrt{2}\\ & |\mathrm{c}{|}^2+|\mathrm{a}{|}^2-2\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{a}}=8\\ & |\mathrm{z}{|}^2+38-12|\mathrm{z}|=8\\ & |\mathrm{z}{|}^2-12|\mathrm{z}|+30=0\\ & |\mathrm{z}|=\frac{12\pm \sqrt{144-120}}{2}\\ & =\frac{12\pm 2\sqrt{6}}{2}\end{array}$ $\begin{array}{rl}& |z|=6+\sqrt{6}\\ & \overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}=\left|\begin{array}{ccc}\hat{\ell }& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 6& 1& -1\\ 1& 1& 0\end{array}\right|\\ & \hat{\ell }-\hat{\mathrm{j}}+5\hat{\mathrm{k}}\\ & |\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|=\sqrt{27}\\ & |(\overrightarrow{\mathrm{a}}\times \mathrm{b})\times \mathrm{z}|=\sqrt{27}(6+\sqrt{6})\frac{\sqrt{3}}{2}\\ & \frac{9}{2}(6+\sqrt{6})\end{array}$