Let R and the three vectors a = i + j + 3 k ,, , b = 2 i + j - k … - Vidyadip

MCQ

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

A
Contains exactly two numbers only one of which is positive
✓
is empty
C
Contains exactly two positive numbers
D
is singleton

✓

Answer

Correct option: B.

is empty

b
$[\vec{a} \cdot \vec{b} \cdot \vec{c}]=0$

$\left| {\begin{array}{*{20}{c}}
\alpha &3&1\\
2&1&{ - \alpha }\\
\alpha &{ - 2}&3
\end{array}} \right| = 0$

$\alpha(3-2 \alpha)+1\left(-\alpha^{2}-6\right)+3(-4-\alpha)=0$

$3 \alpha-2 \alpha^{2}-\alpha^{2}-6-12-3 \alpha=0$

$-3 \alpha^{2}-18=0$

$\alpha^{2}+6=0$ not possible for real $\alpha$

$S$ is empty set

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