Consider the line \(\vec{r} = -2\hat{\imath} + 3\hat{\jmath} + \hat{k} + \lambda(5\hat{\imath} - 3\hat{\jmath} - \hat{k})\). Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | A point on the given line | (I) | \(\left(\displaystyle\frac{5}{\sqrt{35}}, \frac{-3}{\sqrt{35}}, \frac{-1}{\sqrt{35}}\right)\) |
| (B) | Direction ratios of the given line | (II) | (2, 3, 1) |
| (C) | Direction cosines of the given line | (III) | (5, −3, −1) |
| (D) | Direction ratios of a line perpendicular to given line | (IV) | (−2, 3, 1) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
The line \(\vec{r} = \vec{a} + \lambda\vec{b}\) passes through the point with position vector \(\vec{a}\) and is parallel to \(\vec{b}\). Here \(\vec{a} = -2\hat{\imath} + 3\hat{\jmath} + \hat{k}\) and \(\vec{b} = 5\hat{\imath} - 3\hat{\jmath} - \hat{k}\).
(A) A point on the given line
This matches (IV) (−2, 3, 1), the point given by \(\vec{a}\) (\(\lambda = 0\)).
(B) Direction ratios of the given line
This matches (III) (5, −3, −1), the components of \(\vec{b}\).
(C) Direction cosines of the given line
This matches (I): the direction cosines are the direction ratios divided by
\(\sqrt{5^2 + (-3)^2 + (-1)^2} = \sqrt{35}\)
(D) Direction ratios of a line perpendicular to given line
This matches (II) (2, 3, 1), since its dot product with (5, −3, −1) is zero:
\(5(2) + (-3)(3) + (-1)(1) = 10 - 9 - 1 = 0\)
So the matching is (A)-(IV), (B)-(III), (C)-(I), (D)-(II).