Mathematics - Vector Question with Solution | TestHub

$\overrightarrow{\mathrm{a}}=2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+\hat{\mathrm{k}},\overrightarrow{\mathrm{b}}=3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+5\hat{\mathrm{k}}$
$\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}=\left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& -3& 1\\ 3& 2& 5\end{array}\right|$
$\begin{array}{rl}& =-17\hat{i}-7\hat{j}+13\hat{k}\\ & (\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{c}})\times \overrightarrow{\mathrm{b}}=-18\hat{\mathrm{i}}-3\mathrm{j}+12\hat{k}\\ & \Rightarrow (\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})-(\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{b}})=-18\hat{\mathrm{i}}-3\hat{\mathrm{j}}+12\hat{\mathrm{k}}\\ & \Rightarrow \overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}=(-18\hat{\mathrm{i}}-3\hat{\mathrm{j}}+12\hat{k})-(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\\ & =(-18\hat{\mathrm{i}}-3\hat{\mathrm{j}}+12\hat{\mathrm{k}})-(-17\mathrm{i}-7\hat{\mathrm{j}}+13\hat{\mathrm{k}})\\ & \overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}=-\hat{\mathrm{i}}+4\hat{\mathrm{j}}-\hat{\mathrm{k}}\\ & \therefore \overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}}\cdot (\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}})=(2\mathrm{i}-3\mathrm{j}+\mathrm{k})\cdot (-\hat{\mathrm{i}}+4\hat{\mathrm{j}}-\hat{\mathrm{k}})\\ & =-2-12-1=-15\\ & \therefore |\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{d}}|=15\end{array}$