Mathematics - Vector Question with Solution | TestHub

$\begin{array}{rl}& (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{c}}\times (-2\overrightarrow{\mathrm{a}}+3\overrightarrow{\mathrm{b}})=0\\ & (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}+(-2\overrightarrow{\mathrm{a}}+3\overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}=0\\ & \Rightarrow (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})-2\overrightarrow{\mathrm{a}}+3\overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}=0\\ & \Rightarrow \overrightarrow{\mathrm{c}}=\lambda (4\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}})\\ & \Rightarrow =\lambda (44\hat{\mathrm{i}}-4\hat{\mathrm{j}}+4\hat{\mathrm{k}}-4\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})\\ & =\lambda (40\hat{\mathrm{i}}-3\hat{\mathrm{j}}+3\hat{\mathrm{k}})\end{array}$ Now $\begin{array}{rl}& (8\hat{\mathrm{i}}-2\hat{\mathrm{j}}+2\hat{\mathrm{k}}+33\hat{\mathrm{i}}-3\hat{\mathrm{j}}+3\hat{\mathrm{k}})\cdot \lambda (40\hat{\mathrm{i}}-3\hat{\mathrm{j}}+3\hat{\mathrm{k}})=1670\\ & \Rightarrow (41\hat{\mathrm{i}}-5\hat{\mathrm{j}}+5\hat{\mathrm{k}})\cdot (40\hat{\mathrm{i}}-3\hat{\mathrm{j}}+3\hat{\mathrm{k}})\times \lambda =1670)\\ & \Rightarrow (1640+15+15)\lambda =1670\Rightarrow \lambda =1\\ & \text{ so }\overrightarrow{\mathrm{c}}=40\hat{\mathrm{i}}-3\hat{\mathrm{j}}-3\hat{\mathrm{k}}\\ & \Rightarrow |\overrightarrow{\mathrm{c}}{|}^2=1600+9+9=1618\end{array}$