Question

Find $\lambda$ and $\mu$ if ($\hat{i}+3\hat{j}+9\hat{k})\times (3\hat{i}-\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0}$

Answer 1

($\hat{i}+3\hat{j}+9\hat{k})\times (3\hat{i}-\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0}$

$\Rightarrow \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 3& 9\\ 3& -\lambda & \mu \end{array}\right|=0\hat{i}+0\hat{j}+0\hat{k}$

$\Rightarrow (3\mu +9\lambda )\hat{i}-(\mu -27)\hat{j}+(-\lambda -9)\hat{k}=0\hat{i}+0\hat{j}+0\hat{k}$

$\Rightarrow \mu -27=0\text{and}-(\lambda +9)=0$

$\Rightarrow \mu =27\text{and}\lambda =-9$

Hence,
$\lambda =-9$, and $\mu =27$.