Consider the equation of the line $\vec{r} = -\hat{i} + 2\hat{k} + \mu(4\hat{i} - \hat{j} + 2\hat{k})$.

Match List-I with List-II

List-I	List-II
(A) It passes through the point	(I) 4, -1, 2
(B) Its direction ratios are	(II) $\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}$
(C) Its Cartesian form is	(III) (-1, 0, 2)
(D) Its direction cosines are	(IV) $\frac{x+1}{4} = \frac{y}{-1} = \frac{z-2}{2}$

Choose the correct answer from the options given below:

The line is given as: $\vec{r} = -\hat{i} + 2\hat{k} + \mu(4\hat{i} - \hat{j} + 2\hat{k})$

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

Where:

$\vec{a} = -\hat{i} + 2\hat{k}$ represents a point the line passes through

$\vec{b} = 4\hat{i} - \hat{j} + 2\hat{k}$ represents the direction of the line

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(A) It passes through the point → (III) (-1, 0, 2)

From the position vector $\vec{a} = -\hat{i} + 2\hat{k}$:

Coefficient of $\hat{i}$ = -1 → x-coordinate = -1

Coefficient of $\hat{j}$ = 0 → y-coordinate = 0

Coefficient of $\hat{k}$ = 2 → z-coordinate = 2

Point: (-1, 0, 2)

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(B) Its direction ratios are → (I) 4, -1, 2

From $\vec{b} = 4\hat{i} - \hat{j} + 2\hat{k}$:

Coefficient of $\hat{i}$ = 4

Coefficient of $\hat{j}$ = -1

Coefficient of $\hat{k}$ = 2

Direction ratios: 4, -1, 2

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(C) Its Cartesian form is → (IV) $\frac{x+1}{4} = \frac{y}{-1} = \frac{z-2}{2}$

The Cartesian form is: $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$

Where $(x_1, y_1, z_1)$ = (-1, 0, 2) and $(a, b, c)$ = (4, -1, 2)

$\frac{x - (-1)}{4} = \frac{y - 0}{-1} = \frac{z - 2}{2}$

$\frac{x + 1}{4} = \frac{y}{-1} = \frac{z - 2}{2}$

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(D) Its direction cosines are → (II) $\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}$

Direction cosines are obtained by dividing each direction ratio by $\sqrt{a^2 + b^2 + c^2}$

Magnitude:

$\sqrt{4^2 + (-1)^2 + 2^2}$

$= \sqrt{16 + 1 + 4}$

$= \sqrt{21}$

Direction cosines: $\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}$

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Final matching:

(A) → (III) (-1, 0, 2)

(B) → (I) 4, -1, 2

(C) → (IV) $\frac{x+1}{4} = \frac{y}{-1} = \frac{z-2}{2}$

(D) → (II) $\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}$