Match List-I with List-II:
| List-I | List-II | ||
| (A) | \(4\hat{\imath} - 2\hat{\jmath} - 4\hat{k}\) | (I) | A vector perpendicular to both \(\hat{\imath} + 2\hat{\jmath} + \hat{k}\) and \(2\hat{\imath} + 2\hat{\jmath} + 3\hat{k}\) |
| (B) | \(-4\hat{\imath} - 4\hat{\jmath} + 2\hat{k}\) | (II) | Direction ratios are −2, 1, 2 |
| (C) | \(2\hat{\imath} - 4\hat{\jmath} + 4\hat{k}\) | (III) | Angle with the vector \(\hat{\imath} - 2\hat{\jmath} - \hat{k}\) is \(\cos^{-1}\left(\displaystyle\frac{1}{\sqrt{6}}\right)\) |
| (D) | \(4\hat{\imath} - \hat{\jmath} - 2\hat{k}\) | (IV) | Dot product with \(-2\hat{\imath} + \hat{\jmath} + 3\hat{k}\) is 10 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(II), (B)-(IV), (C)-(III), (D)-(I)
(A) \(4\hat{\imath} - 2\hat{\jmath} - 4\hat{k}\)
This matches (II) direction ratios −2, 1, 2.
Its direction ratios 4, −2, −4 are proportional to −2, 1, 2:
\((4, -2, -4) = -2\,(-2, 1, 2)\)
(B) \(-4\hat{\imath} - 4\hat{\jmath} + 2\hat{k}\)
This matches (IV) dot product 10.
\((-4)(-2) + (-4)(1) + (2)(3) = 8 - 4 + 6 = 10\)
(C) \(2\hat{\imath} - 4\hat{\jmath} + 4\hat{k}\)
This matches (III) angle \(\cos^{-1}\frac{1}{\sqrt{6}}\) with \(\hat{\imath} - 2\hat{\jmath} - \hat{k}\).
\(\text{dot product} = 2 + 8 - 4 = 6\)
\(\text{magnitudes} = \sqrt{4 + 16 + 16} = 6 \text{ and } \sqrt{1 + 4 + 1} = \sqrt{6}\)
\( \displaystyle \cos\theta = \frac{6}{6\sqrt{6}} = \frac{1}{\sqrt{6}} \)
(D) \(4\hat{\imath} - \hat{\jmath} - 2\hat{k}\)
This matches (I) a vector perpendicular to both given vectors.
The cross product is perpendicular to both:
\( \displaystyle \left(\hat{\imath} + 2\hat{\jmath} + \hat{k}\right)\times\left(2\hat{\imath} + 2\hat{\jmath} + 3\hat{k}\right) = \begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\1 & 2 & 1\\2 & 2 & 3\end{vmatrix} \)
\(= (6 - 2)\hat{\imath} - (3 - 2)\hat{\jmath} + (2 - 4)\hat{k}\)
\(= 4\hat{\imath} - \hat{\jmath} - 2\hat{k}\)
So the matching is (A)-(II), (B)-(IV), (C)-(III), (D)-(I).