Question

The number of distinct real values of $\lambda$ for which vectors $-{\lambda }^2\hat{i}+\hat{j}+\hat{k},\hat{i}-{\lambda }^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-{\lambda }^2\hat{k}$ are coplanar is

Answer 1

For the given vectors to be coplanar
$\Rightarrow \left|\begin{array}{ccc}-{\lambda }^2& 1& 1\\ 1& -{\lambda }^2& 1\\ 1& 1& -{\lambda }^2\end{array}\right|=0$
$\Rightarrow -{\lambda }^2({\lambda }^4-1)-1(-{\lambda }^2-1)+1(1+{\lambda }^2)=0$
$\Rightarrow {\lambda }^6-3{\lambda }^2-2=0$
$\Rightarrow (1+{\lambda }^2{)}^2({\lambda }^2-2)$
$\Rightarrow \lambda =\pm \sqrt{2}$