Step 1: Conceptual Foundation:
The vector form of a line is expressed as $\vec{r} = \vec{a} + \lambda\vec{b}$. Here, $\vec{a}$ represents the position vector of a point situated on the line, while $\vec{b}$ is a vector parallel to the line. The components of $\vec{b}$ are the direction ratios.
Step 3: Detailed Analysis:
The provided equation is $\vec{r} = (\hat{i} - 2\hat{j} + 4\hat{k}) + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k})$.
Comparing this with the standard form $\vec{r} = \vec{a} + \lambda\vec{b}$, we identify:
$\vec{a} = \hat{i} - 2\hat{j} + 4\hat{k}$
$\vec{b} = -\hat{i} + 2\hat{j} - 4\hat{k}$
(A) Point on the Line:
The position vector $\vec{a}$ denotes a point on the line with coordinates (1, -2, 4). This corresponds to option (III).
(B) Direction Ratios:
The components of the parallel vector $\vec{b}$ are the direction ratios of the line, which are (-1, 2, -4). This corresponds to option (IV).
(C) Direction Cosines:
Direction cosines are derived from the unit vector along $\vec{b}$. First, calculate the magnitude of $\vec{b}$.
$|\vec{b}| = \sqrt{(-1)^2 + 2^2 + (-4)^2} = \sqrt{1 + 4 + 16} = \sqrt{21}$.
The unit vector is $\hat{b} = \frac{\vec{b}}{|\vec{b}|} = \frac{-\hat{i} + 2\hat{j} - 4\hat{k}}{\sqrt{21}}$.
The direction cosines are $(-\frac{1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, -\frac{4}{\sqrt{21}})$. This corresponds to option (I).
(D) Direction Ratios of a Perpendicular Line:
Let the direction ratios of a perpendicular line be $(l, m, n)$. The dot product of its direction vector and $\vec{b}$ must be zero.
$\vec{b} \cdot (l\hat{i} + m\hat{j} + n\hat{k}) = 0$
$(-1)(l) + (2)(m) + (-4)(n) = 0 \implies -l + 2m - 4n = 0$.
We test option (II), (4, -2, -2), for these direction ratios.
$-(4) + 2(-2) - 4(-2) = -4 - 4 + 8 = 0$.
The condition is satisfied, meaning (4, -2, -2) are the direction ratios of a perpendicular line. This corresponds to option (II).
Step 4: Conclusion:
The correct pairings are (A) - (III), (B) - (IV), (C) - (I), and (D) - (II). This aligns with option (3).