Match List-I with List-II

List-I	List-II
Equation of line	Information
(A) $\vec{r} = (3\hat{i} - 2\hat{j} + \hat{k}) + \lambda(\hat{j} - 2\hat{k})$	(I) Direction ratios are 2, 4, -1
(B) $\frac{2-x}{1} = \frac{2y+1}{4}$, $z = 2$	(II) Perpendicular to $2\hat{i} - \hat{j} + \hat{k}$
(C) $\frac{x}{1} = \frac{y-3}{2} = \frac{3-4z}{2}$	(III) Passing through the point $(3, -2, 1)$
(D) $\vec{r} = (3\hat{i} + 2\hat{j} + \hat{k}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$	(IV) Direction ratios are -1, 2, 0

Choose the correct answer from the options given below:

For vector form of a line $\vec{r} = \vec{a} + \lambda\vec{b}$, the vector $\vec{a}$ gives a point on the line and $\vec{b}$ gives the direction ratios.

For Cartesian form $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$, the direction ratios are $a, b, c$.

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For (A): $\vec{r} = (3\hat{i} - 2\hat{j} + \hat{k}) + \lambda(\hat{j} - 2\hat{k})$

Position vector $= 3\hat{i} - 2\hat{j} + \hat{k}$

The line passes through point $(3, -2, 1)$

Direction vector $= 0\hat{i} + 1\hat{j} - 2\hat{k}$

Direction ratios: $(0, 1, -2)$

(A) matches with (III)

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For (B): $\frac{2-x}{1} = \frac{2y+1}{4}$, $z = 2$

Rewriting: $\frac{2-x}{1} = \frac{x-2}{-1}$

$\frac{2y+1}{4} = \frac{y+\frac{1}{2}}{2}$

Since $z = 2$ (constant), the direction ratio for $z$ is $0$

Direction ratios: $(-1, 2, 0)$

(B) matches with (IV)

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For (C): $\frac{x}{1} = \frac{y-3}{2} = \frac{3-4z}{2}$

Rewriting the z-term:

$\frac{3-4z}{2}$

$= \frac{-(4z-3)}{2}$

$= \frac{4z-3}{-2}$

Direction ratios from the equation: $(1, 2, -\frac{1}{2})$

Multiplying by $2$: $(2, 4, -1)$

(C) matches with (I)

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For (D): $\vec{r} = (3\hat{i} + 2\hat{j} + \hat{k}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$

Direction vector of line D $= 2\hat{i} + \hat{j} - 3\hat{k}$

Two vectors are perpendicular if their dot product $= 0$

Dot product with $2\hat{i} - \hat{j} + \hat{k}$:

$(2)(2) + (1)(-1) + (-3)(1)$

$= 4 - 1 - 3$

$= 0$

The line is perpendicular to $2\hat{i} - \hat{j} + \hat{k}$

(D) matches with (II)

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