Question

$\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-4\hat{k},\overrightarrow{c}=\hat{i}+\lambda \hat{j}+3\hat{k}$ are coplanar, then the value of $\lambda$ is

• A. $\frac{5}{2}$
• B. $\frac{3}{5}$
• C. $\frac{5}{3}$
• D. None of these

Answer 1

Answer: (C)

$\frac{5}{3}$

If given vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are coplanar, then
$\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]=0$
$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ 2& 0& -4\\ 1& \lambda & 3\end{array}\right|=0$
$\Rightarrow 1(0+4\lambda )-1(6+4)+1(2\lambda -0)=0$
$\Rightarrow 4\lambda -10+2\lambda =0$
$\Rightarrow 6\lambda =10$
$\Rightarrow \lambda =\frac{5}{3}$