Question

If $\overline{a},\overline{b},\overline{c}$ are non-coplanar vectors and $\overline{\alpha }=\overline{b}\times \overline{c},\overline{\beta }=\overline{c}\times \overline{a},\overline{\gamma }=\overline{a}\times \overline{b}$, then which of these is true

• A. $\overline{\alpha }$ is parallel to $\overline{b}$
• B. $\overline{\alpha }$ is parallel to $\overline{c}$
• C. $\overline{\alpha }$ is perpendicular to $\overline{b}$
• D. $\overline{\gamma }$ is parallel to $\overline{a}$

Answer 1

Answer: (C)

$\overline{\alpha }$ is perpendicular to $\overline{b}$

$\overline{\alpha }\cdot \overline{b}=(\overline{b}\times \overline{c})\cdot \overline{b}=0$
$\Rightarrow \overline{\alpha }$ is perpendicular to $\overline{b}$

$\overline{\alpha }\cdot \overline{c}=(\overline{b}\times \overline{c})\cdot \overline{c}=0$
$\Rightarrow \overline{\alpha }$ is perpendicular to $\overline{c}$

$\overline{\gamma }\cdot \overline{b}=(\overline{a}\times \overline{b})\cdot \overline{a}=0$
$\Rightarrow \overline{\gamma }$ is perpendicular to $\overline{a}$

Hence, only option 'B' is correct