Match List-I with List-II
| List-I (pair of vectors) | List-II (relation) | ||
| (A) | \(2\hat{\imath}-\hat{\jmath}+3\hat{k}\) and \(-4\hat{\imath}+2\hat{\jmath}-6\hat{k}\) | (I) | collinear, in the same direction |
| (B) | \(\hat{\imath}+2\hat{\jmath}-\hat{k}\) and \(3\hat{\imath}-\hat{\jmath}+\hat{k}\) | (II) | collinear, in opposite directions |
| (C) | \(\hat{\imath}-3\hat{\jmath}+2\hat{k}\) and \(2\hat{\imath}-6\hat{\jmath}+4\hat{k}\) | (III) | perpendicular |
| (D) | \(2\hat{\imath}+\hat{\jmath}+\hat{k}\) and \(\hat{\imath}+2\hat{\jmath}+\hat{k}\) | (IV) | neither collinear nor perpendicular |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
Two vectors are collinear when one is a scalar multiple of the other: a positive multiple gives the same direction, a negative multiple the opposite direction. They are perpendicular when their scalar product is zero.
(A) \(2\hat{\imath}-\hat{\jmath}+3\hat{k}\) and \(-4\hat{\imath}+2\hat{\jmath}-6\hat{k}\)
This matches (II) collinear, in opposite directions.
\(-4\hat{\imath}+2\hat{\jmath}-6\hat{k} = -2(2\hat{\imath}-\hat{\jmath}+3\hat{k})\)
(B) \(\hat{\imath}+2\hat{\jmath}-\hat{k}\) and \(3\hat{\imath}-\hat{\jmath}+\hat{k}\)
This matches (III) perpendicular.
\((1)(3)+(2)(-1)+(-1)(1) = 3-2-1 = 0\)
(C) \(\hat{\imath}-3\hat{\jmath}+2\hat{k}\) and \(2\hat{\imath}-6\hat{\jmath}+4\hat{k}\)
This matches (I) collinear, in the same direction.
\(2\hat{\imath}-6\hat{\jmath}+4\hat{k} = 2(\hat{\imath}-3\hat{\jmath}+2\hat{k})\)
(D) \(2\hat{\imath}+\hat{\jmath}+\hat{k}\) and \(\hat{\imath}+2\hat{\jmath}+\hat{k}\)
This matches (IV) neither collinear nor perpendicular.
\((2)(1)+(1)(2)+(1)(1) = 5 \ne 0\)
The components 2, 1, 1 and 1, 2, 1 are not in proportion, so the vectors are not collinear either.