Question

Let $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}-2\hat{j}+3\hat{k}$ Then the value of $\lambda$ for which the vector $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+(2\lambda -1)\hat{k}$ is parallel to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$ is

• A. 1
• B. 0
• C. -1
• D. 2

Answer 1

The correct option is B 0

As $\overline{c}$ is parallel to palne containing $\overline{a}$ and $\overline{b}$
i.e $\overline{c}$ is perpendicular to $\overline{a}\times \overline{b}$
$\Rightarrow \left[\overline{c}\overline{a}\overline{b}\right]=0$
$\Rightarrow \left|\begin{array}{ccc}\lambda & 1& 2\lambda -1\\ 2& 3& -1\\ 1& -2& 3\end{array}\right|=0$
$\Rightarrow 7\lambda -7-14\lambda +7=0$
$\therefore \lambda =0$
Hence, option B.