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Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVectors2 Marks
MCQ
The vectors $\lambda \hat{i}+\hat{j}+2 \hat{k}, \hat{i}+\lambda \hat{j}-\hat{k}$ and $2 \hat{ i }-\hat{ j }+\lambda \hat{ k }$ are coplanar if
  • ✓
    $\lambda=-2$
  • B
    $\lambda=0$
  • C
    $\lambda=2$
  • D
    $\lambda=1$
✓

Answer

Correct option: A.
$\lambda=-2$
(A) Let $\overline{ a }, \overline{ b }$ and $\overline{ c }$ be the given vectors.
The given vectors are coplanar.
$\therefore\left|\begin{array}{ccc}\lambda & 1 & 2 \\ 1 & \lambda & -1 \\ 2 & -1 & \lambda\end{array}\right|=0$
$\begin{array}{l}\Rightarrow \lambda\left(\lambda^2-1\right)-(\lambda+2)+2(-1-2 \lambda)=0 \\ \Rightarrow \lambda^3-6 \lambda-4=0 \\ \Rightarrow(\lambda+2)\left(\lambda^2-2 \lambda-2\right)=0\end{array}$
$\Rightarrow \lambda=-2$ or $\lambda=\frac{2 \pm \sqrt{4+8}}{2}=1 \pm \sqrt{3}$

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