Match List-I with List-II

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

Choose the correct answer from the options given below:

To find the angle between two vectors, the dot product formula is used:

$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \times |\vec{b}|}$

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

$\vec{a} \cdot \vec{b} = (1)(1) + (-1)(1) + (0)(0)$

$= 1 - 1$

$= 0$

$|\vec{a}| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$

$|\vec{b}| = \sqrt{1^2 + 1^2} = \sqrt{2}$

$\cos \theta = \frac{0}{\sqrt{2} \times \sqrt{2}}$

$= 0$

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

(A) matches with (IV)

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

The second vector is exactly the negative of the first, pointing in opposite directions.

$\vec{a} \cdot \vec{b} = (1)(-1) + (-1)(1) + (1)(-1)$

$= -1 - 1 - 1$

$= -3$

$|\vec{a}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$

$|\vec{b}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$

$\cos \theta = \frac{-3}{\sqrt{3} \times \sqrt{3}}$

$= \frac{-3}{3}$

$= -1$

$\theta = \pi$

(B) matches with (I)

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

$\vec{a} \cdot \vec{b} = (1)(-1) + (1)(0)$

$= -1$

$|\vec{a}| = \sqrt{1^2 + 1^2} = \sqrt{2}$

$|\vec{b}| = \sqrt{(-1)^2} = 1$

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

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

$= -\frac{\sqrt{2}}{2}$

$\theta = \frac{3\pi}{4}$

(C) matches with (II)

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

$\vec{a} \cdot \vec{b} = (1)(0) + (0)(0) + (1)(1)$

$= 1$

$|\vec{a}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}$

$|\vec{b}| = \sqrt{0^2 + 0^2 + 1^2} = 1$

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

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

$= \frac{\sqrt{2}}{2}$

$\theta = \frac{\pi}{4}$

(D) matches with (III)

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Final Matching:

(A) → (IV) $\frac{\pi}{2}$

(B) → (I) $\pi$

(C) → (II) $\frac{3\pi}{4}$

(D) → (III) $\frac{\pi}{4}$