Question

The vectors \(λ \widehat i + \widehat j + 2\widehat k\), \(\widehat i + λ \widehat j - \widehat k\) and \(2\widehat i - \widehat j + λ \widehat k\) are coplanar if λ =
1. -2
2. 0
3. 1
4. -1

Answer 1

Correct Answer - Option 1 : -2

Concept used:

a, b, and c vectors are coplanar when \([\vec{a}, \vec{b},\vec{c}] = 0\)

\(\left| {\begin{array}{*{20}{c}} {λ}&{b_1}&{c_1}\\ {a_2}&{λ }&{c_2}\\ {a_3}&{b_3}&{λ } \end{array}} \right| = 0\)

Calculation:

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

\(\Rightarrow λ(λ^2 - 1) - 1(λ +2) + 2(-1 - 2λ) = 0\)

\(\Rightarrow λ^3 - λ - λ - 2 - 2 - 4λ = 0\)

\(\Rightarrow λ^3 - 6λ -4 = 0\)

\(\Rightarrow (λ + 2)(λ^2 -2λ -2) = 0\)

\(\Rightarrow λ = -2 \ or\ λ= \frac{2±\sqrt{12}}{2}\)

∴ λ = - 2