Consider \(\overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}\) and \(\overrightarrow{\mathrm{c}}\) are non-zero vectors such that \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{a}}=0\), \(|\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{r}}||\overrightarrow{\mathrm{b}}|,|\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{c}}|=|\overrightarrow{\mathrm{r}}||\overrightarrow{\mathrm{c}}|\), then \([\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]\) is

Here, it is given in problem \(|\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{r}}||\overrightarrow{\mathrm{b}}|\)
So, it is clear that angle between \(\overrightarrow{\mathrm{r}}\) and \(\overrightarrow{\mathrm{b}}\) is \(\frac{\pi}{2}\) \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{a}}=0 \Rightarrow \overrightarrow{\mathrm{r}}\) is perpendicular to \(\overrightarrow{\mathrm{a}}\). \(|\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{c}}|=|\overrightarrow{\mathrm{r}}| \cdot|\overrightarrow{\mathrm{c}}| \Rightarrow \overrightarrow{\mathrm{r}}\) is perpendicular to \(\overrightarrow{\mathrm{c}}\). Thus, \(\overrightarrow{\mathrm{r}}\) is perpendicular to \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}\) and \(\overrightarrow{\mathrm{c}}\).
Hence, \(\vec{a}, \vec{b}\) and \(\vec{c}\) are coplanar.
\(\Rightarrow[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]=0\)