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

Match List-I with List-II

List-I	List-II
(A) A point on the given line	(I) $\left(\frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, \frac{-4}{\sqrt{21}}\right)$
(B) direction ratios of the line	(II) $(4, -2, -2)$
(C) direction cosines of the line	(III) $(1, -2, 4)$
(D) direction ratios of a line perpendicular to given line	(IV) $(-1, 2, -4)$

Choose the correct answer from the options given below:

Given line: $\vec{r} = \hat{i} - 2\hat{j} + 4\hat{k} + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k})$

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

Where:

$\vec{a} = \hat{i} - 2\hat{j} + 4\hat{k}$ (position vector of a point on the line)

$\vec{b} = -\hat{i} + 2\hat{j} - 4\hat{k}$ (direction vector)

---

(A) A point on the given line

Setting $\lambda = 0$:

$\vec{r} = \hat{i} - 2\hat{j} + 4\hat{k}$

This gives the point $(1, -2, 4)$

(A) → (III)

---

(B) Direction ratios of the line

The direction vector is $-\hat{i} + 2\hat{j} - 4\hat{k}$

Direction ratios are the coefficients: $(-1, 2, -4)$

(B) → (IV)

---

(C) Direction cosines of the line

Direction cosines are the normalized direction ratios.

Magnitude of direction vector:

$|\vec{b}| = \sqrt{(-1)^2 + (2)^2 + (-4)^2}$

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

$= \sqrt{21}$

Direction cosines:

$\left(\frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, \frac{-4}{\sqrt{21}}\right)$

(C) → (I)

---

(D) Direction ratios of a line perpendicular to given line

Two lines are perpendicular when their dot product equals zero.

Given line has direction ratios: $(-1, 2, -4)$

Testing $(4, -2, -2)$:

$(-1)(4) + (2)(-2) + (-4)(-2)$

$= -4 - 4 + 8$

$= 0$

The lines are perpendicular.

(D) → (II)

---

The correct matching is: (A) → (III), (B) → (IV), (C) → (I), (D) → (II)