Match List-I with List-II

Choose the correct answer from the options given below:

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

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

(A) If $\vec{a} = \lambda \vec{b}$ and $|\vec{a}| = |\vec{b}|$

$\Rightarrow |\lambda| = 1 \Rightarrow \lambda = \pm 1 \Rightarrow \vec{a} = \pm \vec{b}$

⟹ (A) → (III)

(B) Projection vector of $\vec{a}$ on $\vec{b}$ is $\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$

⟹ (B) → (IV)

(C) $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \Rightarrow \vec{a} \cdot \vec{b} = 0$

⟹ $\vec{a}$ and $\vec{b}$ are orthogonal

⟹ (C) → (I)

(D) $|\lambda \vec{a}| = |\lambda||\vec{a}| = 4|\lambda|$, where $\lambda \in [-3, 2]$

⟹ $|\lambda| \in [0, 3] \Rightarrow |\lambda \vec{a}| \in [0, 12]$

⟹ (D) → (II)