If $\hat{p}, \vec{q}, \vec{r}$ are non-coplanar vectors such that $\hat{p} \times \vec{q} = \vec{r}, \vec{q} \times \vec{r} = \hat{p}$ then Consider following statements
(i) $| \hat{p} | = | \vec{q} | = | \vec{r} |$
(ii) $\hat{p}, \vec{q}, \vec{r}$ are mutually perpendicular vectors
(iii) Volume of parallelepiped formed by co-initial edges $\hat{p}, \vec{q}, \vec{r}$ is $[\hat{p} \vec{q} \vec{r}]^{2}$
(iv) $| \hat{p} | + | \vec{q} | + | \vec{r} | = 3$
Number of INCORRECT statements are

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Detailed Solution

$\begin{array}{l} \hat{p} \times \vec{q} = \vec{r} and \vec{q} \times \vec{r} = \hat{p} \\ \Rightarrow (\vec{q} \times \vec{r}) \times \vec{q} = \vec{r} \\ | q |^{2} \vec{r} - (\vec{r} \cdot \vec{q}) \vec{q} = \vec{r} \\ \Rightarrow | q | = 1 and \vec{r} \cdot \vec{q} = 0 \end{array}$
$Similarly \vec{q} \times (\hat{p} \times \vec{q}) = \hat{p}$
$\begin{array}{l} | q | = 1 and \vec{p} \cdot \vec{q} = 0 and [\hat{p} \vec{q} \vec{r}] = | \vec{r} |^{2} = | \hat{p} |^{2} \\ \Rightarrow | \hat{p} | = | \vec{q} | = | \vec{r} | = 1 \\ \vec{r} \cdot \hat{p} = 0 = \vec{r} \cdot \vec{q} = \vec{p} \cdot \vec{q} = 0 \end{array}$
$\Rightarrow$ Mutually perpendicular
$\begin{aligned} \Rightarrow | \hat{p} | = | \vec{q} | = | \vec{r} | = 1 \\ \Rightarrow [\hat{p} \vec{q} \vec{r}] = 1 = [\hat{p} \vec{q} \vec{r}]^{2} \end{aligned}$