Question

Let $\alpha \in R$ and the three vectors $\overrightarrow{a}=\alpha \hat{i}+\hat{j}+3\hat{k},\overrightarrow{b}=2\hat{i}+\hat{j}-\alpha \hat{k}$ and $\overrightarrow{c}=\alpha \hat{i}-2\hat{j}+3\hat{k}$. Then the set $S=\{\alpha :\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{are coplanar}\}$

• A. is empty
• B. is singleton
• C. contains exactly two positive numbers
• D. contains exactly two numbers only one of which is positive

Answer 1

The correct option is A is empty

For these vectors to be coplanar,
$\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=0\Rightarrow \left|\begin{array}{ccc}\alpha & 1& 3\\ 2& 1& -\alpha \\ \alpha & -2& 3\end{array}\right|=0$
$R_1\to R_1-R_3\left|\begin{array}{ccc}0& 3& 0\\ 2& 1& -\alpha \\ \alpha & -2& 3\end{array}\right|=0$
$\Rightarrow -3(6+{\alpha }^2)=0$
$\Rightarrow {\alpha }^2=-6$
Not possible for real $\alpha$
So, $S$ is empty.