Question

Find $λ$ and $μ$ if $(2\hat{i}+6\hat{j}+7\hat{k})\times(\hat{i}+λ\hat{j}+μ\hat{k})=\vec{0}.$

Answer 1

$(2\hat{i}+6\hat{j}+7\hat{k})\times(\hat{i}+λ\hat{j}+μ\hat{k})=\vec{0}.$
$⇒\begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\\ 2& 6 & 27 \\\1& \lambda& µ\end{vmatrix}|=0\hat{i}+0\hat{j}+0\hat{k}$
$⇒\hat{i}(6μ-27\lambda)-\hat{j}(2μ-27)+\hat{k}(27λ-6)=0\hat{i}+0\hat{j}+0\hat{k}$
On comparing the corresponding components,we have:
$6μ-27λ=0$
$2μ-27=0$
$2λ-6=0$
Now,
$2λ-6=0⇒λ=3$
$2μ-27=0⇒μ=\frac{27}{2}$
$Hence,λ=3 \space and \space μ=\frac{27}{2}.$