Question

If $\overrightarrow{a}=\hat{i}-2\hat{j}+3\hat{k}$, $\overrightarrow{b}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+(2\lambda -1)\hat{k}$ are coplanar vectors, then $\lambda$ is equal to?

• A. $1$
• B. $2$
• C. $-1$
• D. $0$

Answer 1

Answer: (D)

$0$

Given $\overrightarrow{a}=\hat{i}-2\hat{j}+3\hat{k}$, $\overrightarrow{b}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+(2\lambda -1)\hat{k}$ are coplanar

The vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar if,

$\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=0$

$\Rightarrow \left|\begin{array}{ccc}1& -2& 3\\ 2& 3& -1\\ \lambda & 1& 2\lambda -1\end{array}\right|=0$

$\Rightarrow 1(6\lambda -3+1)+2(4\lambda -2+\lambda )+3(2-3\lambda )=0$

$\Rightarrow 6\lambda -2+2(5\lambda -2)+3(2-3\lambda )=0$

$\Rightarrow 7\lambda =0$

$\Rightarrow \lambda =0$