Question

If $\overline{p},\overline{q}$ and $\overline{r}$ are non zero, non coplanar vectors then $[\overline{p}+\overline{q}-\overline{r}\overline{p}-\overline{q}\overline{q}-\overline{r}]=$ _________.

• A. $3\left[\overline{p}\overline{q}\overline{r}\right]$
• B. $0$
• C. $\left[\overline{p}\overline{q}\overline{r}\right]$
• D. $2\left[\overline{p}\overline{q}\overline{r}\right]$

Answer 1

The correct option is C $\left[\overline{p}\overline{q}\overline{r}\right]$

$[\overrightarrow{p}+\overrightarrow{q}-\overrightarrow{r}\overrightarrow{p}-\overrightarrow{q}\overrightarrow{q}-\overrightarrow{r}]$

$=(\overrightarrow{p}+\overrightarrow{q}-\overrightarrow{r})\times (\overrightarrow{p}-\overrightarrow{q}).(\overrightarrow{q}-\overrightarrow{r})$

$=(-2\overrightarrow{p}\times \overrightarrow{q}-\overrightarrow{r}\times \overrightarrow{p}+\overrightarrow{r}\times \overrightarrow{q}).(\overrightarrow{q}-\overrightarrow{r})$

$=\overrightarrow{p}\times \overrightarrow{q}.\overrightarrow{r}$

$=\left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{r}\right]$