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

 

Answers (1)

\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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