Browse Questions

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

$\begin{array}{1 1} 0 \\ 1 \\ 2 \\ 3 \end{array}$

For three vectors to be coplanar the condition is
$\left|\begin{array}{ccc} -\lambda^2 & 1 &1\\1 &-\lambda^2 & 1\\1 & 1& -\lambda^2\end {array}\right |=0$
Solving which $-\lambda^2(\lambda^4-1)-(-\lambda^2-1)+(1+\lambda^2)=0$
$\Rightarrow\:(\lambda^2+1)^2(2-\lambda^2)=0$
$\Rightarrow\:\lambda=\pm \sqrt 2$
$\therefore$ There are two real values of $\lambda$.