Question

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

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

Hint:

For a line $\vec{r} = \vec{a} + \lambda\vec{b}$:

The components of $\vec{a}$ give a point on the line.
The components of $\vec{b}$ give the direction ratios.
The components of the unit vector $\hat{b}$ give the direction cosines.
Any vector whose dot product with $\vec{b}$ is zero is perpendicular to the line.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The vector equation of a line is given in the form $\vec{r} = \vec{a} + \lambda\vec{b}$, where $\vec{a}$ is the position vector of a point on the line and $\vec{b}$ is a vector parallel to the line, whose components give the direction ratios.
Step 3: Detailed Explanation:
The given equation is $\vec{r} = (\hat{i} - 2\hat{j} + 4\hat{k}) + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k})$.
By comparing with $\vec{r} = \vec{a} + \lambda\vec{b}$, we have:
$\vec{a} = \hat{i} - 2\hat{j} + 4\hat{k}$
$\vec{b} = -\hat{i} + 2\hat{j} - 4\hat{k}$
(A) A point on the given line:
The position vector $\vec{a}$ corresponds to a point on the line. The coordinates of this point are (1, -2, 4). This matches with (III).
(B) Direction ratios of the line:
The components of the parallel vector $\vec{b}$ are the direction ratios of the line. These are (-1, 2, -4). This matches with (IV).
(C) Direction cosines of the line:
Direction cosines are the components of the unit vector in the direction of $\vec{b}$. First, find 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 matches with (I).
(D) Direction ratios of a line perpendicular to given line:
Let the direction ratios of a perpendicular line be $(l, m, n)$. The dot product of its direction vector with $\vec{b}$ 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 need to check which direction ratios from List-II satisfy this condition. Let's test option (II) (4, -2, -2).
$-(4) + 2(-2) - 4(-2) = -4 - 4 + 8 = 0$.
The condition is satisfied. So, (4, -2, -2) are direction ratios of a perpendicular line. This matches with (II).
Step 4: Final Answer:
The correct matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II). This corresponds to option (3).