MHT CET · Maths · Vector Algebra
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 \(-1\)
- D \(-8\)
Answer & Solution
Correct Answer
(D) \(-8\)
Step-by-step Solution
Detailed explanation
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 \([\mathbf{a} \mathbf{b} \mathbf{c}]=0\)
\(
\begin{array}{l}
\Rightarrow \left|\begin{array}{ccc}
2 & -1 & -1 \\
1 & 2 & -3 \\
3 & \lambda & 5
\end{array}\right|=0 \\
\Rightarrow 2(10+3 \lambda)+(5+9)-(\lambda-6)=0 \\
\Rightarrow 20+6 \lambda+14-\lambda+6=0 \\
\Rightarrow 5 \lambda+40=0 \\
\Rightarrow \lambda=-8
\end{array}
\)
\(
\mathbf{b}=\mathbf{i}+2 \mathbf{j}-3 \mathbf{k}
\)
and
\(
\mathbf{c}=3 \mathbf{i}+\lambda \mathbf{j}+5 \mathbf{k}
\)
If these vectors are coplanar, then \([\mathbf{a} \mathbf{b} \mathbf{c}]=0\)
\(
\begin{array}{l}
\Rightarrow \left|\begin{array}{ccc}
2 & -1 & -1 \\
1 & 2 & -3 \\
3 & \lambda & 5
\end{array}\right|=0 \\
\Rightarrow 2(10+3 \lambda)+(5+9)-(\lambda-6)=0 \\
\Rightarrow 20+6 \lambda+14-\lambda+6=0 \\
\Rightarrow 5 \lambda+40=0 \\
\Rightarrow \lambda=-8
\end{array}
\)
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