Question

Let α ∈ R and three vectors \(\vec a = \alpha \hat i + \hat j + 3\hat k,\;\vec b = 2\hat i + \hat j - \alpha \hat k,{\rm{\;and\;}}\vec c = \alpha \hat i - 2\hat j + 3\hat k\). Then the set \(S = \left\{ {\alpha :\vec a,\;\vec b{\rm{\;and\;}}\vec c{\rm{\;are\;coplanar}}} \right\}\)
1. Is singleton
2. Is empty
3. Contains exactly two positive numbers
4. Contains exactly two numbers only one of which is positive

Answer 1

Correct Answer - Option 2 : Is empty

From question, the vectors given are:

\(\vec a = \alpha \hat i + \hat j + 3\hat k\)

\(\vec b = 2\hat i + \hat j - \alpha \hat k\)

\(\vec c = \alpha \hat i - 2\hat j + 3\hat k\)

From question, the set has three given vectors which are coplanar.

The condition of coplanar is:

\(\left[ {\vec a\vec b\vec c} \right] = 0\)

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

⇒ α(3 – 2α) -1(6 + α²) + 3(-4 – α) = 0

⇒ 3α – 2α² – 6 – α² – 12 – 3α = 0

⇒ -3α² = 18

⇒ α² = -6

Which is a imaginary number.

∴ S = ϕ

Thus, the set ‘S’ is an empty set.