Q55 / 150 · MHT CET 2012 · Mathematics
If three vectors $2 \mathbf{i}-\mathbf{j}-\mathbf{k}, \mathbf{i}+2 \mathbf{j}-3 \mathbf{k}$ and
$3 \mathbf{i}+\lambda \mathbf{j}+5 \mathbf{k}$ are coplanar, then the value of $\lambda$ is
- A$-4$
- B$-2$
- C$-1$
- Correct answerD$-8$
Solution
Let $\quad \mathbf{a}=2 \mathbf{i}-\mathbf{j}-\mathbf{k}$,
$$
\mathbf{b}=\mathbf{i}+2 \mathbf{j}-3 \mathbf{k}
$$
and
$$
\mathbf{c}=3 \mathbf{i}+\lambda \mathbf{j}+5 \mathbf{k}
$$
If these vectors are coplanar, then $[\mathbf{a} \mathbf{b} \mathbf{c}]=0$
$$
\begin{array}{l}
\Rightarrow \quad\left|\begin{array}{ccc}
2 & -1 & -1 \\
1 & 2 & -3 \\
3 & \lambda & 5
\end{array}\right|=0 \\
\Rightarrow \quad 2(10+3 \lambda)+(5+9)-(\lambda-6)=0 \\
\Rightarrow 20+6 \lambda+14-\lambda+6=0 \\
\Rightarrow 5 \lambda+40=0 \\
\Rightarrow \lambda=-8
\end{array}
$$
Hence, the correct answer is option: 'D'.