Question

If $\overrightarrow{x}$ is a vector such that $\left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{r}\right]\overrightarrow{x}=(\overrightarrow{p}\times \overrightarrow{q})\times \overrightarrow{r},$ then $\overrightarrow{x}$ is

• A. $c(\hat{i}-2\hat{j}+\hat{k})$
• B. a unit vector
• C. indeterminate, as $\left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{r}\right]$
• D. $-\frac{1}{2}(\hat{i}-2\hat{j}+\hat{k})$

Answer 1

The correct option is C indeterminate, as $\left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{r}\right]$

Let $p=\overline{i}$

$q=\overline{j}$
$r=\overline{k}$

Hence

$\overline{p}\times \overline{q}$
$=k$

Now

$\left[\right(\overline{p}\times \overline{q}).\overline{k}].\overline{x}=(\overline{p}\times \overline{q})\times \overline{r}$
$[\overline{k}.\overline{k}]\overline{x}=(\overline{k}\times \overline{k})$

Thus vector (x) turns out to be 0.

Hence the value of vector x is determined by the chosen values of vectors, p, q, and r.

Unless the value of p,q, and r are not fixed, we cannot quantify x.

Hence the correct answer is 'indeterminate'.