Question

If $a+2b-c=0$ and
$a\times b+b\times c+c\times a=\lambda (a\times b)$, then the value of $\lambda$ is equal to

• A. $5$
• B. $4$
• C. $2$
• D. $-2$
• E. $-4$

Answer 1

The correct option is D $-2$

Given, $a\times b+b\times c+c\times a=\lambda (a\times b)...\left(i\right)$
and $a+2b-c=0$
Let $a=\hat{i};b=\hat{j}\Rightarrow c=\hat{i}+2\hat{j}$
Substituting the values in Eq(i)
$\left(\hat{i}\right)\times \left(\hat{j}\right)+\left(\hat{j}\right)(\hat{i}+2\hat{j})+(\hat{i}+2\hat{j})\times \hat{i}=\lambda \left(\hat{i}\right)\times \left(\hat{j}\right)$
$\Rightarrow \left(\hat{k}\right)+(-\hat{k})+-2\left(\hat{k}\right)=\lambda \left(\hat{k}\right)$
$\Rightarrow \lambda =-2$