Question

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

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

Answer 1

Answer: (D)

$-8$

Let $a=2\hat{i}-\hat{j}-\hat{k}$,
$b=\hat{i}+2\hat{j}-3\hat{k}$
and $c=3\hat{i}+\lambda \hat{j}+5\hat{k}$

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