Question

The vectors $\lambda i+j+2k,i+\lambda j-k$ and $2i-j+\lambda k$ are coplanar if

• A. $\lambda =-2$
• B. $\lambda =1+\sqrt{3}$
• C. $\lambda =1-\sqrt{3}$
• D. $\lambda =0$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $\lambda =-2$
B $\lambda =1+\sqrt{3}$
C $\lambda =1-\sqrt{3}$

The given vectors are coplanar if and only if
$\left|\begin{array}{ccc}\lambda & 1& 2\\ 1& \lambda & -1\\ 2& -1& \lambda \end{array}\right|=0$
$\Rightarrow \lambda ({\lambda }^2-1)-(\lambda +2)+2(-1-2\lambda )=0$
$\Rightarrow {\lambda }^3-6\lambda -4=0$
$\Rightarrow (\lambda +2)({\lambda }^2-2\lambda -2)=0$
$\lambda =-2$ or $\lambda =\frac{2\pm \sqrt{4+8}}{2}=1\pm \sqrt{3}$