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

Match List-I with List-II

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

Choose the correct answer from the options given below:

The correct option is: (A) - (III), (B) - (IV), (C) - (I), (D) - (II).

Let's analyze the given vector equation of the line step-by-step and match the items of List-I with List-II.

The equation of the line is given by:
$$\overrightarrow{r}=(\hat{i}-2\hat{j}+4\hat{k})+\lambda (-\hat{i}+2\hat{j}-4\hat{k})$$

This is in the standard vector form of a line:
$$\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$$
where:
- $\overrightarrow{a}=\hat{i}-2\hat{j}+4\hat{k}$ represents the position vector of a point on the line.
- $\overrightarrow{b}=-\hat{i}+2\hat{j}-4\hat{k}$ represents the direction vector parallel to the line.

Step 1: Match (A) - A point on the given line
From the position vector $\overrightarrow{a}=1\hat{i}-2\hat{j}+4\hat{k}$, the coordinates of a point on the line are:
$(1,-2,4)$
This matches with item (III) in List-II. Thus, (A) - (III).

Step 2: Match (B) - Direction ratios of the line
The components of the direction vector $\overrightarrow{b}=-1\hat{i}+2\hat{j}-4\hat{k}$ give the direction ratios of the line:
$(-1,2,-4)$
This matches with item (IV) in List-II. Thus, (B) - (IV).

Step 3: Match (C) - Direction cosines of the line
To find the direction cosines, we normalize the direction ratios $(a,b,c)=(-1,2,-4)$ by dividing each component by the magnitude of the direction vector:
$$\text{Magnitude}=\sqrt{{(-1)}^2+2^2+{(-4)}^2}=\sqrt{1+4+16}=\sqrt{21}$$
Therefore, the direction cosines are:
$$(\frac{-1}{\sqrt{21}},\frac{2}{\sqrt{21}},\frac{-4}{\sqrt{21}})$$
This matches with item (I) in List-II. Thus, (C) - (I).

Step 4: Match (D) - Direction ratios of a line perpendicular to the given line
Let the direction ratios of a perpendicular line be $(a_1,b_1,c_1)$. For two lines to be perpendicular, the dot product of their direction vectors must be zero:
$$a_1{a}_2+b_1{b}_2+c_1{c}_2=0$$
Substituting the direction ratios of our line $(a_2,b_2,c_2)=(-1,2,-4)$:
$$-1\left(a_1\right)+2\left(b_1\right)-4\left(c_1\right)=0$$
Let's test the remaining item from List-II, which is (II) (4, -2, -2):
$$-1\left(4\right)+2(-2)-4(-2)=-4-4+8=0$$
Since the dot product is exactly $0$, a line with direction ratios $(4,-2,-2)$ is perpendicular to the given line. Thus, (D) - (II).