Match List-I with List-II:

Choose the correct answer from the options given below:

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

The correct answer is Option (2) → (A)-(II), (B)-(IV), (C)-(III), (D)-(I)

(A) $4\hat i-2\hat j-4\hat k$ have d.c. as -

$\frac{4}{\sqrt{36}}\hat i-\frac{2}{\sqrt{36}}\hat j-\frac{4}{\sqrt{36}}\hat k$

$\frac{2}{3}\hat i-\frac{1}{3}\hat j-\frac{2}{3}\hat k$

∴ D.r. is → 2, -1, 2 or -2, 1, 2

(B) $(-4\hat i-4\hat j + 2\hat k).(-2\hat i+\hat j+3\hat k)=8-4+6$

$=10$

(C) $\cos θ=\frac{2(1)+(-4)(-2)+(4)(-1)}{\sqrt{36}\sqrt{6}}=\frac{2+8-4}{6\sqrt{6}}=\frac{1}{\sqrt{6}}$

(D) $\begin{vmatrix}\hat i&\hat j&\hat k\\1&2&1\\2&2&3\end{vmatrix}=4\hat i-\hat j-2\hat k$

∴ This is perpendicular to both of them.