The vectors \(3\hat{\imath}-\hat{\jmath}+2\hat{k}\), \(2\hat{\imath}+\hat{\jmath}+3\hat{k}\) and \(\hat{\imath}+\lambda\hat{\jmath}-\hat{k}\) are coplanar if λ is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → −2
Three vectors are coplanar exactly when their scalar triple product is zero, that is, when the determinant of their components is zero:
\( \displaystyle \begin{vmatrix} 3 & -1 & 2 \\ 2 & 1 & 3 \\ 1 & \lambda & -1 \end{vmatrix} = 0 \)
Expanding along the first row:
\(3\left[(1)(-1) - 3\lambda\right] + 1\left[(2)(-1) - (3)(1)\right] + 2\left[2\lambda - 1\right] = 0\)
\(3(-1 - 3\lambda) + (-5) + (4\lambda - 2) = 0\)
\(-3 - 9\lambda - 5 + 4\lambda - 2 = 0\)
\(-10 - 5\lambda = 0\)
\(\lambda = -2\)
Note: This topic was not in the CUET 2026 syllabus.