# Let $\alpha \epsilon R$ and the three vectors$\vec{a}=\alpha\hat{i}+\hat{j}+3\hat{k}$ , $\vec{a}=2\hat{i}+\hat{j}-\alpha \hat{k}$ and $\vec{a}=\alpha \hat{i}-2\hat{j}+3 \hat{k}$ . Then the set $S=\left \{{\alpha: \vec{a},\vec{b}\: \: and \: \: \vec{c}\: \: are\: \: coplanar }}{ \right \}$ Option 1) is singleton Option 2) is empty Option 3) contains exactly two positive numbers Option 4) contains exactly two numbers only one of which is positive

$\begin{vmatrix} \alpha & 1 & 3\\ 2 & 1 & -4\\ \alpha & -2 &-3 \end{vmatrix}=0$

=>  $3\alpha ^{2}+18=0$

=>  $\alpha \epsilon \phi$

Option 1)

is singleton

Option 2)

is empty

Option 3)

contains exactly two positive numbers

Option 4)

contains exactly two numbers only one of which is positive

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