Match List-I with List-II List-I List-II (A) Angle between hat i-hat…

The correct answer is Option (1) → (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

List-I	List-II
(A) Angle between $\hat i-\hat j$ and $\hat i+\hat j$	(IV) $\frac{\pi}{2}$
(B) Angle between $\hat i-\hat j+\hat k$ and $-\hat i+\hat j-\hat k$	(I) $\pi$
(C) Angle between $\hat i+\hat j$ and $-\hat i$	(II) $\frac{3\pi}{4}$
(D) Angle between $\hat i+\hat k$ and $\hat k$	(III) $\frac{\pi}{4}$

(A) Angle between $\hat i-\hat j$ and $\hat i+\hat j$

Dot product:

$(1,-1,0)\cdot(1,1,0)=1-1=0$

Angle $=\frac{\pi}{2}$ → (IV)

(B) Angle between $\hat i-\hat j+\hat k$ and $-\hat i+\hat j-\hat k$

Dot product:

$(1,-1,1)\cdot(-1,1,-1)=-1-1-1=-3$

Magnitudes $=\sqrt{3},\sqrt{3}$

$\cos\theta=\frac{-3}{3}=-1$

Angle $=\pi$ → (I)

(C) Angle between $\hat i+\hat j$ and $-\hat i$

Dot product:

$(1,1,0)\cdot(-1,0,0)=-1$

Magnitudes $=\sqrt{2},1$

$\cos\theta=\frac{-1}{\sqrt{2}}$

Angle $=\frac{3\pi}{4}$ → (II)

(D) Angle between $\hat i+\hat k$ and $\hat k$

Dot product:

$(1,0,1)\cdot(0,0,1)=1$

Magnitudes $=\sqrt{2},1$

$\cos\theta=\frac{1}{\sqrt{2}}$

Angle $=\frac{\pi}{4}$ → (III)

Correct matching: A–IV, B–I, C–II, D–III