Consider the line vec r=hat i-2hat j+4hat k+λ(-hat i+2hat j-4hat k) M…

The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

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

Given line

$\vec r=\hat i-2\hat j+4\hat k+\lambda(-\hat i+2\hat j-4\hat k)$

A point on the line is obtained by taking $\lambda=0$

Point $=(1,-2,4)$

$(A)\rightarrow(III)$

Direction vector of the line

$\vec d=(-1,2,-4)$

Hence direction ratios are $(-1,2,-4)$

$(B)\rightarrow(IV)$

Magnitude of direction vector

$|\vec d|=\sqrt{(-1)^2+2^2+(-4)^2}=\sqrt{21}$

Direction cosines

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

$(C)\rightarrow(I)$

A line perpendicular to the given line must have direction ratios proportional to $(4,-2,-2)$ since

$(-1,2,-4)\cdot(4,-2,-2)=-4-4+8=0$

$(D)\rightarrow(II)$

Final Matching: (A)-(III), (B)-(IV), (C)-(I), (D)-(II).