Question

If $\overrightarrow{r}·\overrightarrow{a}=\overrightarrow{r}·\overrightarrow{b}=\overrightarrow{r}·\overrightarrow{c}=0$ for some non-zero vector $\overrightarrow{r},$ then the value of $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right],$ is
(a) 2
(b) 3
(c) 0
(d) none of these

Answer 1

(c) 0

$$\begin{gathered} \mathrm{If}\overrightarrow{r}\mathit{.}\overrightarrow{a}=0\mathrm{for}\mathrm{some}\mathrm{non}-\mathrm{zero}\mathrm{vector}\overrightarrow{r},\mathrm{then}\mathrm{either}\overrightarrow{\mathrm{a}}\mathrm{is}\mathrm{a}\mathrm{zero}-\mathrm{vector}\mathrm{or}\mathrm{it}\mathrm{is}\mathrm{perpendicular}\mathrm{to}\overrightarrow{r}. \\ \mathrm{If}\mathrm{one}\mathrm{of}\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}},\overline{\mathrm{c}}\mathrm{is}\mathrm{zero},\mathrm{then}\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\mathit{=}\mathit{0}\mathit{} \\ \mathrm{If}\mathrm{all}\overrightarrow{a}\mathit{,}\mathit{}\overrightarrow{b}\mathit{}\mathrm{and}\mathit{}\overrightarrow{c}\mathrm{are}\mathrm{non}-\mathrm{zero},\mathrm{then}\mathrm{they}\mathrm{must}\mathrm{be}\mathrm{coplanar}\mathrm{as}\mathrm{they}\mathrm{are}\mathrm{perpendicular}\mathrm{to}\mathrm{vector}\overrightarrow{r}\mathit{.} \\ \therefore \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\mathit{=}0 \end{gathered}$$