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

List-I	List-II
(A) A point on the given line	(I) $\left(\frac{5}{\sqrt{35}}, \frac{-3}{\sqrt{35}}, \frac{-1}{\sqrt{35}}\right)$
(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:

The line is given as: $\vec{r} = -2\hat{i} + 3\hat{j} + \hat{k} + \lambda(5\hat{i} - 3\hat{j} - \hat{k})$

This is in the vector form $\vec{r} = \vec{a} + \lambda\vec{b}$ where:

$\vec{a} = -2\hat{i} + 3\hat{j} + \hat{k}$ gives a point on the line

$\vec{b} = 5\hat{i} - 3\hat{j} - \hat{k}$ gives the direction of the line

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(A) A point on the given line → (IV) (-2, 3, 1)

Setting $\lambda = 0$:

$\vec{r} = -2\hat{i} + 3\hat{j} + \hat{k} + 0(5\hat{i} - 3\hat{j} - \hat{k})$

$\vec{r} = -2\hat{i} + 3\hat{j} + \hat{k}$

The point is $(-2, 3, 1)$

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(B) Direction ratios of the given line → (III) (5, -3, -1)

From the direction vector $\vec{b} = 5\hat{i} - 3\hat{j} - \hat{k}$

Direction ratios are $(5, -3, -1)$

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(C) Direction cosines of the given line → (I) $\left(\frac{5}{\sqrt{35}}, \frac{-3}{\sqrt{35}}, \frac{-1}{\sqrt{35}}\right)$

Direction cosines are obtained by dividing direction ratios by their magnitude.

For direction ratios $(a, b, c)$:

Direction cosines $= \left(\frac{a}{\sqrt{a^2+b^2+c^2}}, \frac{b}{\sqrt{a^2+b^2+c^2}}, \frac{c}{\sqrt{a^2+b^2+c^2}}\right)$

Magnitude $= \sqrt{5^2 + (-3)^2 + (-1)^2}$

$= \sqrt{25 + 9 + 1}$

$= \sqrt{35}$

Direction cosines $= \left(\frac{5}{\sqrt{35}}, \frac{-3}{\sqrt{35}}, \frac{-1}{\sqrt{35}}\right)$

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(D) Direction ratios of a line perpendicular to given line → (II) (2, 3, 1)

Two lines are perpendicular when their direction vectors have dot product equal to zero.

For the given line with direction ratios $(5, -3, -1)$ and perpendicular line with $(a, b, c)$:

$5a - 3b - c = 0$

Checking option (II): $(2, 3, 1)$

$5(2) - 3(3) - 1 = 10 - 9 - 1 = 0$

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Final Matching:

(A) → (IV) $(-2, 3, 1)$

(B) → (III) $(5, -3, -1)$

(C) → (I) $\left(\frac{5}{\sqrt{35}}, \frac{-3}{\sqrt{35}}, \frac{-1}{\sqrt{35}}\right)$

(D) → (II) $(2, 3, 1)$