• # question_answer If $[\overset{\to }{\mathop{p}}\,+2\overset{\to }{\mathop{q}}\,+3\overset{\to }{\mathop{r}}\,\overset{\to }{\mathop{q}}\,+2\overset{\to }{\mathop{r}}\,+3\overset{\to }{\mathop{p}}\,\,\overset{\to }{\mathop{r}}\,+2\overset{\to }{\mathop{p}}\,+3\overset{\to }{\mathop{q}}\,]=54,$where $\overset{\to }{\mathop{p}}\,,\overset{\to }{\mathop{q}}\,$ and $\overset{\to }{\mathop{r}}\,$ are three vectors then the value of$\left| \begin{matrix} \overset{\to }{\mathop{p}}\,.\overset{\to }{\mathop{p}}\, & \overset{\to }{\mathop{p}}\,.\overset{\to }{\mathop{q}}\, & \overset{\to }{\mathop{p}}\,.\overset{\to }{\mathop{r}}\, \\ \overset{\to }{\mathop{p}}\,.\overset{\to }{\mathop{q}}\, & \overset{\to }{\mathop{q}}\,.\overset{\to }{\mathop{q}}\, & \overset{\to }{\mathop{q}}\,.\overset{\to }{\mathop{r}}\, \\ \overset{\to }{\mathop{p}}\,.\overset{\to }{\mathop{r}}\, & \overset{\to }{\mathop{r}}\,.\overset{\to }{\mathop{q}}\, & \overset{\to }{\mathop{r}}\,.\overset{\to }{\mathop{r}}\, \\ \end{matrix} \right|$ is A)  4                                             B)  9 C)  16                                          D)  36

Solution :

$(\overrightarrow{p}+2\overrightarrow{q}+3\overrightarrow{r})\cdot \left( (\overrightarrow{q}+2\overrightarrow{r}+3\overrightarrow{p})\times (\overrightarrow{r}+2\overrightarrow{p}+3\overrightarrow{q}) \right)=54$On solving $18\left[ \overrightarrow{p}\,\,\,\overrightarrow{q}\,\,\,\overrightarrow{r} \right]=54$$\left[ \overrightarrow{p}\,\,\,\overrightarrow{q}\,\,\,\overrightarrow{r} \right]=3$ Hence,${{\left[ \overrightarrow{p}\,\,\,\overrightarrow{q}\,\,\,\overrightarrow{r} \right]}^{2}}=9$Ans.

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