Consider the line $\vec r = -2\hat i+3\hat j+\hat k+λ(5\hat i-3\hat j-\hat k)$. Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(IV), (B)-(III), (C)-(I), (D)-(II)

The correct answer is Option (2) → (A)-(IV), (B)-(III), (C)-(I), (D)-(II)

Given Line: \(\vec{r} = -2\hat{i} + 3\hat{j} + \hat{k} + \lambda(5\hat{i} - 3\hat{j} - \hat{k})\)

List-I and List-II Matching Using Definitions:

(A) A point on the given line: From the vector equation, the fixed point is the position vector: → Point is \( (-2, 3, 1) \), which matches with (IV).

(B) Direction ratios of the given line: Direction vector is the coefficient of λ, i.e., \( (5, -3, -1) \), which matches with (III).

(C) Direction cosines of the given line: These are the direction ratios divided by their magnitude. → Magnitude = \( \sqrt{5^2 + (-3)^2 + (-1)^2} = \sqrt{25 + 9 + 1} = \sqrt{35} \) → Direction cosines = \( \left(\frac{5}{\sqrt{35}}, \frac{-3}{\sqrt{35}}, \frac{-1}{\sqrt{35}}\right) \), which matches with (I).

(D) Direction ratios of a line perpendicular to the given line: Let required direction ratios be \( (a, b, c) \). Then perpendicular condition: \( 5a - 3b - c = 0 \). One such triple satisfying this is \( (2, 3, 1) \), which matches with (II).