Question

If $l(\overline{b}\times \overline{c})+m(\overline{c}\times \overline{a})+n(\overline{a}\times \overline{b})=0$And at least one of $l,m,n$ is not zero, then the vectors $\overline{a},\overline{b},\overline{c}$ are

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

Answer 1

Answer: (B)

Coplanar

Given that,

$l\left(\overline{b}\times \overline{c}\right)+m\left(\overline{c}\times \overline{a}\right)+n\left(\overline{a}\times \overline{b}\right)=0$

Taking Dot product with $\overline{a};l\left[\overline{a}.\overline{b}.\overline{c}\right]=0$

Dot product with $\overline{b};m\left[\overline{a}.\overline{b}.\overline{c}\right]=0$

Dot product with $\overline{c};n\left[\overline{a}.\overline{b}.\overline{c}\right]=0$

$\Rightarrow \left[\overline{a}.\overline{b}.\overline{c}\right]=0$

$\Rightarrow \overline{a}.\overline{b}$ and $\overline{c}$ are co-planar.

Then,

OPtion $B$ is correct answer.