Question

If $a(\overrightarrow{\alpha }\times \overrightarrow{\beta })+b(\overrightarrow{\beta }\times \overrightarrow{\gamma })+c(\overrightarrow{\gamma }\times \overrightarrow{\alpha })=\overrightarrow{0}$ and atleast one of $a,b,c$ are non-zero then the vectors $\overrightarrow{\alpha },\overrightarrow{\beta },\overrightarrow{\gamma }$ are

• A. Coplanar
• B. Mutually perpendicular
• C. Parallel
• D. None of these

Answer 1

The correct option is A Coplanar

Let $C$ be Non Zero, then
$\left(a\right(\overline{\alpha }\times \overline{\beta })+b(\overline{\beta }\times \overline{\gamma })+c(\overline{\gamma }\times \overline{\alpha }\left)\right)=0$
$=c\left[\overline{\gamma }\overline{\alpha }\overline{\beta }\right]=0$
So $\left[\overline{\gamma }\overline{\alpha }\overline{\beta }\right]=0$

So $\overline{\gamma }\times \overline{\alpha }\perp \overline{\beta }$
So $\beta ||\overline{\gamma }$ & $\overline{\beta }||\overline{\alpha }$

$\beta =k\overline{\gamma }+\gamma \overline{\alpha }$