Question

If three vectors $2i-j-k$ , $i+2j-3k$ and $3i+\lambda j+5k$ are coplanar, then the value of $\lambda$ is

• A. $-4$
• B. $-2$
• C. $-1$
• D. $-8$

Answer 1

Answer: (D)

Let $a=2i-j-k$,
$b=i+2j-3k$
and $c=3i+\lambda j+5k$
If these vectors are coplanar, then $[abc]=0$
$\Rightarrow \left|\begin{array}{ccc}2& -1& -1\\ 1& 2& -3\\ 3& \lambda & 5\end{array}\right|=0$
$\Rightarrow 2(10+3\lambda )+(5+9)-(\lambda -6)=0$
$\Rightarrow 20+6\lambda +14-\lambda +6=0$
$\Rightarrow 5\lambda +40=0$
$\Rightarrow \lambda =-8$