KCET · Maths · Three Dimensional Geometry
If the vectors \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) are coplanar, then the value of \(\lambda\) is
- A 6
- B \(-5\)
- C -6
- D 5
Answer & Solution
Correct Answer
(A) 6
Step-by-step Solution
Detailed explanation
Given vectors \(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) are coplanar
\(\left|\begin{array}{ccc}2 & -3 & 4 \\ 2 & 1 & -1 \\ \lambda & -1 & 2\end{array}\right|=0\)
\(\Rightarrow \quad 2(2-1)+3(4+\lambda)+4(-2-\lambda)=0\)
\(\Rightarrow \quad 2+12+3 \lambda-8-4 \lambda=0\)
\(\Rightarrow \quad 6-\lambda=0\)
\(\Rightarrow \quad \lambda=6\)
\(\left|\begin{array}{ccc}2 & -3 & 4 \\ 2 & 1 & -1 \\ \lambda & -1 & 2\end{array}\right|=0\)
\(\Rightarrow \quad 2(2-1)+3(4+\lambda)+4(-2-\lambda)=0\)
\(\Rightarrow \quad 2+12+3 \lambda-8-4 \lambda=0\)
\(\Rightarrow \quad 6-\lambda=0\)
\(\Rightarrow \quad \lambda=6\)
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