Match List-I with List-II

Choose the correct answer from the options given below:

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

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

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

Dot product

$(\hat i-\hat j)\cdot(\hat j+\hat k)=0-1+0=-1$

Magnitudes

$|\hat i-\hat j|=\sqrt2,\;|\hat j+\hat k|=\sqrt2$

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

$\theta=\frac{2\pi}{3}$

(A) $\rightarrow$ (II)

(B) Angle between $2\hat j-\hat k$ and $\hat j+2\hat k$

Dot product

$(2\hat j-\hat k)\cdot(\hat j+2\hat k)=2-2=0$

Hence angle $=\frac{\pi}{2}$

(B) $\rightarrow$ (IV)

(C) Angle between $\hat i+2\hat j$ and $5\hat i+10\hat j$

Second vector is scalar multiple of first

Angle $=0$

(C) $\rightarrow$ (I)

(D) Angle between $\sqrt3\hat i+\hat j$ and $\hat i$

Dot product

$(\sqrt3\hat i+\hat j)\cdot\hat i=\sqrt3$

Magnitudes

$|\sqrt3\hat i+\hat j|=2,\;|\hat i|=1$

$\cos\theta=\frac{\sqrt3}{2}$

$\theta=\frac{\pi}{6}$

(D) $\rightarrow$ (III)

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